Private neighbors, perfect codes and their relation with the $\vt$-number of closed neighborhood ideals

PRIV A TE NEIGHBORS, PERFECT CODES AND THEIR RELA TION WITH THE v -NUMBER OF CLOSED NEIGHBORHOOD IDEALS DELIO JARAMILLO-VELEZ, HIRAM H. L ´ OPEZ, AND R ODRIGO SAN-JOS ´ E Abstract. In this w ork, w e in v estigate the connections betw een dominating sets, priv ate neigh- b ors, and p erfect co des in graphs, and their relationships with commutativ e algebra. In partic- ular, w e estimate the v -num b er of closed neighborho o d ideals in terms of minimal dominating sets and priv ate neighbors. W e show ho w the v -num b er is related to other graph inv arian ts, suc h as the co ver n umber, domination n umber, and matc hing num b er. Moreov er, we explore the relation with the Castelnuo vo-Mumford regularity , proving that the v -num b er is a low er b ound for the regularity of bipartite and w ell-co vered graphs. Finally , dra wing from the relation b e- t ween efficient dominating set and p erfect codes, we use the redundancy of Hamming co des to presen t lo wer and upp er b ounds for the v -num b er of some sp ecial family of graphs. 1. Introduction A dominating set in a graph is a subset of v ertices D such that any other vertex of the graph has a neigh b or in D . A large part of the interest in dominating sets comes from its wide range of connections across man y areas, including chemistry , computer communication net works, facilit y lo cation, so cial netw orks, surv eying, monitoring electrical p o wer netw orks, genetics, co ding theory , and sev eral other branches of mathematics; see [ HHH23 ]. As an in tro- ductory example of these connections, consider a graph represen ting a communication netw ork in whic h information tra vels along the edges. Then, assume that transmitters are placed on a set of v ertices D and that u is an external priv ate neigh b or of D , meaning that u has only one neigh b or v in D . Then, removing v from D w ould directly affect the net w ork’s communication b ecause no other transmitter in D \ { v } w ould reac h u . In this sense, w e say that D minimally dominates the communication of the netw ork [ Co c99 ]. The previous example illustrates the imp ortance of dominating sets and also in tro duces the relev ant concept of priv ate neighbors. These vertices are such that make the set D minimal with respec t to the prop erty of domina- tion [ CHM78 , FFHJ94 , HR25 ]. In this w ork, we sho w relationships b et ween dominating sets, priv ate neighbors, perfect co des of a graph, and commutativ e algebra. W e do this by analyzing the v -n umber of closed neighborho o d ideals. The V asconcelos n umber, or just v -n umber, is an algebraic in v arian t of a graded ideal I in a p olynomial ring S := K [ t 1 , . . . , t n ] o ver a field K , defined as (1) v ( I ) := min { d ≥ 0 | ∃ f ∈ S d and p ∈ Ass( I ) with ( I : f ) = p } , 2020 Mathematics Subje ct Classific ation. Primary 05E40, 05C69; Secondary 94B05. Key wor ds and phr ases. Closed neighborho o d ideal, v -n umber, Castelnuo vo-Mumford regularit y , priv ate neigh- b ors, dominating set, p erfect co de. Delio Jaramillo-V elez was partially supp orted by the NSF grant DMS-2401558. Hiram H. L´ op ez was partially supp orted b y the Commonw ealth Cyb er Initiative and by the NSF grants DMS-2401558 and DMS-2502705. Ro drigo San-Jos´ e was partially supp orted by the Commonw ealth Cyb er Initiative, the NSF grant DMS-2401558, and b y Gran t PID2022-137283NB-C22 funded b y MICIU/AEI/10.13039/501100011033 and b y ERDF/EU. 1 2 JARAMILLO-VELEZ, L ´ OPEZ, AND SAN-JOS ´ E where S d denotes the d -th graded component of S , Ass( I ) denotes the set of associated primes of I , and ( I : f ) is the colon ideal. The v -n umber w as introduced by Coop er, Secelean u, T oh˘ anean u, V az Pinto, and Villarreal in connection with the study of the asymptotic b eha vior of the mini- m um distance of pro jective Reed–Muller-t yp e co des [ CST + 20 , Vil26 ]. This algebraic in v ariant is tied to indicator functions, whic h arise in co ding theory [ LSV21 , Sor91 ], Cayley–Bac harac h sc hemes [ GKR93 ], and interpolation problems [ KR00 ]. The v -num b er has serv ed as a ric h source of algebraic interpretations of graph inv arian ts thanks to ideals asso ciated with graphs. One of the most well-kno wn of these ideals is the edge ideal I ( G ), which is essen tially generated b y the edges of the graph G ; see [ Vil90 ]. He re, the v -n umber of I ( G ) corresp onds to the minimal cardinality of an indep endence set of G such that its neighbors are a minimal v ertex cov er [ JV21 , GR V21 ]. Suc h a characterization provides algebraic conditions to deriv e conclusions ab out the graph [ JV21 ]; for example, an algebraic classification of W 2 graphs. F or the case of binomial edge ideals, the v -num b er at the first minimal prime corresp onds with the connected domination num b er [ JVS24 , ASS24 ], giving thus a relation b etw een the v -num b er and the concept of domination. Motiv ated b y the relations b et ween comm utative algebra and graph theory , in this pap er, w e establish a com binatorial expression of the v -n um b er of closed neigh b orho o d ideals; see Theo- rem 3.6 . The closed neighborho o d ideal of a graph G is generated by the closed neigh b orho o ds of the vertices of G . These ideals can b e viewed as a generalization of edge ideals, but instead of considering only one neighbor p er vertex, we consider its en tire neighborho o d [ SM20 ]. Recen tly , neigh b orho o d ideals ha ve attracted considerable attention, and researchers hav e presen ted sev- eral com binatorial-algebraic results that include an expression for the primary decomp osition, whic h dep ends on the minimal dominating sets [ HSW22 ]. Moreo ver, there has b een consider- able in terest in the combinatorial estimation of homological inv arian ts of neighborho o d ideals, including the Castelnuo v o–Mumford regularit y and the pro jective dimension [ MS26 , CJRS25 ]. In Section 2 , we introduce notation and preliminary results regarding the main sub jects of this w ork: graph theory , commutativ e algebra, and p erfect co des. W e refer the reader to T able 1 for a summary of the notation w e use in this w ork. Section 3 is devoted to presenting a combinatorial expression for the v -num b er of closed neigh b orho o d ideals, which corresp onds to Theorem 3.6 . This expression is purely in terms of minimal dominating sets and their asso ciated priv ate neighbors. In Section 4 , we present relations b etw een the v -n um b er and sev eral other graph inv arian ts suc h as the cov er num b er, the domination num b er, and the matching num b er; see Theorem 4.1 . W e address the relation b et w een the v -num b er and the C asteln uov o-Mumford regularity . W e sho w that the v -n umber is a lo w er b ound for sev eral families of graphs, and conjecture the general case; see Theorem 4.7 and Conjecture 4.8 . This is motiv ated by the now classical case of edge ideals, where the v -num b er can b e arbitrarily greater than the regularit y [ Civ23 ]. In Section 5 , we consider p erfect codes in the v ector space F m q , where F q is a finite field. A co de C ⊆ F m q is e -p erfect if for ev ery element v ∈ F m q there is a unique element w in C such that w and v differ in at most e en tries. By representing the v ector space F m q as a sp ecific graph Γ( m, q ), the problem of existence of e -p erfect co des corresp ondences with finding sp ecial domination sets, called efficient dominating sets, with the prop erty that the neighborho o ds of their elemen ts form a partition of the graph [ HR25 , Big73 ]. The problem of the existence of e -p erfect co des w as solved by Tiet¨ av¨ ainen, who pro ved that the only nontrivial p erfect co des are the 1-error-correcting Hamming co des together with t wo exceptional co des first disco v ered b y Golay [ Tie73 ]. W e asso ciate the v -num b er to co ding theory by presenting lo w er and upp er PRIV A TE NEIGHBORS, PERFECT CODES, v -NUMBER, AND CLOSED NEIGHBORHOOD IDEALS 3 b ounds for the v -num b er of the closed neigh b orho o d ideal N Γ( q r − 1 q − 1 ,q ) in terms of the redundancy of the existent Hamming code; see Theorem 5.5 . In App endix A , w e pro vide the Sage co de used for implemen tations of the closed neigh b orho o d ideals, v -n umber form ula, and computations of the Casteln uov o-Mumford regularit y [ The25 ]. 2. Preliminaries In this section, we presen t terminology and preliminary results. W e divide the section in to three parts: graph theory , comm utative algebra, and co ding theory . Additionally , T able 1 summarizes all of the notation. 2.1. Graph theory. A gr aph G is a pair ( V ( G ) , E ( G )), where V ( G ) is a finite set and E ( G ) = {{ u, v } | u, v ∈ V ( G ) } . The elemen ts of V ( G ) and E ( G ) are called vertices and edges of G , resp ectively . W e consider only simple gr aphs , i.e., graphs without multiple edges or lo ops. W e say that a graph is c onne cte d if for ev ery pair of vertices there is a path b et ween them. A graph H is a subgraph of G if V ( H ) ⊆ V ( G ) and E ( H ) ⊆ E ( G ). A subgraph H of G is called induc e d if E ( H ) = {{ u, v } ∈ E ( G ) | u, v ∈ V ( H ) } . F or an edge e = { u, v } ∈ E ( G ), we say that e is incident to u and v , u is adjacen t to v , or u is a neighbor of v . F or a vertex v ∈ V ( G ), w e define: • The neighb orho o d of v , denoted b y N ( v ), is the set of all neighbors of v . • The close d neighb orho o d of v is giv en by N [ v ] := N ( v ) ∪ { v } . Similarly , for a set of vertices U ⊆ V ( G ), w e define: • The neighb orho o d of U is N ( U ) := S v ∈ U N ( v ) . • The close d neighb orho o d of U is N [ U ] := N ( U ) ∪ U. 2.1.1. Private neighb ors. The concept of priv ate neigh b ors in graphs w as in tro duced b y Co ck- a yne, Hedetniemi, and Miller in 1978 [ CHM78 ]. Let w ∈ V ( G ) b e a v ertex and U ⊂ V ( G ) a set of v ertices in a graph G . Assume that w is a private neighb or of U , meaning that | N ( w ) ∩ U | = 1. • If w / ∈ U , then w is called an external private neighb or of U . • If w ∈ U and deg( w ) = 1 in the induced graph G [ U ], then w is called an internal private neighb or of U ; see [ HR25 ]. • If w ∈ U and deg( w ) = 0 in the induced graph G [ U ], then w is called a self-private neighb or of U ; see [ HR25 ]. W e use P N ( U ) to denote the set of all external and self-private neighb ors of vertices U . 2.1.2. Dominating and irr e dundant sets. W e now introduce the concept of an irredundant set of a graph, whic h is related to priv ate neighbors. Let D and U be tw o sets of v ertices of a graph G . W e say that D dominates U if U ⊆ N [ D ] . The set D is a dominating set of G if N [ D ] = V ( G ). The domination numb er of G , denoted b y γ ( G ), is the minimum cardinalit y of a dominating set of G . A dominating set is called minimal if it do es not prop erly con tain another dominating set. Note that if a dominating set D is 4 JARAMILLO-VELEZ, L ´ OPEZ, AND SAN-JOS ´ E minimal, then each vertex of D has a priv ate neighbor. Therefore, if w e remov e an y vertex from D , the resulting set is no longer a dominating set. A reference for dominating sets is [ HHH23 ]. Definition 2.1. A set of vertices D ⊂ V ( G ) is an irr e dundant set of G if for ev ery u ∈ D there is v ∈ V ( G ) suc h that N [ v ] ∩ D = { u } . In w ords, a set D is irredundan t if and only if every v ertex u ∈ D has at least one priv ate neigh b or in P N ( D ): either itself, when N [ u ] ∩ D = { u } , or a vertex v ∈ V ( G ) \ D suc h that N ( v ) ∩ D = { u } . An irredundant set D is said to b e maximal if it is not prop erly con tained in another irre- dundan t set. Let D ⊆ V ( G ) b e a set of vertices and M ⊆ E ( G ) a set of edges of a graph G . • W e say that D is a vertex c over of G if every edge of G is incident with at least one v ertex in D . A vertex cov er is minimal if it is minimal with resp ect to inclusion. The vertex c over numb er of G , denoted by τ ( G ), is the minimum cardinalit y of a v ertex cov er of G . • W e say that D is an indep endent set of G if no tw o vertices in D are adjacent. An indep enden t set is maximal if it is maximal with resp ect to inclusion. The indep endenc e numb er of G , denoted b y i ( G ), is the maxim um cardinality of an indep endent set of G . • W e sa y that M is a matching of G if no tw o edges in M hav e a common endp oin t. The matching numb er of G , denoted b y a ( G ), is the maxim um cardinalit y of a matching of G . The follo wing result, which can be found in [ CLZ11 , Cor. 10.37], relates minimal dominating sets with maximal indep enden t sets. Prop osition 2.2. Every maximal indep endent set is a minimal dominating set. Remark 2.3. As a consequence of Prop osition 2.2 , we alwa ys ha ve γ ( G ) ≤ i ( G ). 2.2. Comm utative algebra. W e denote the p olynomial ring o ver a field K with the standard grading b y S := K [ t 1 , . . . , t n ] = ∞ M d =0 S d . Let I b e a graded ideal of S . A prime ideal p of S is an asso ciate d prime of S/I if ( I : f ) := p for some f ∈ S d , where ( I : f ) = { g ∈ S | g f ∈ I } is the colon ideal b et w een I and f . The set of asso ciated primes of S/I is denoted b y Ass( I ). In this work, we are interested in the relation b etw een the v -num b er and the Castelnuo vo- Mumford regularity , which is defined as follows. Consider the minimal graded free resolution of S/I as an S -mo dule: 0 → M j S ( − j ) b g,j → · · · → M j S ( − j ) b 1 ,j → S → S/I → 0 . The Castelnuovo–Mumfor d r e gularity of S /I , or simply the r e gularity of S/I , is defined as reg ( S/I ) := max { j − i | b i,j  = 0 } . PRIV A TE NEIGHBORS, PERFECT CODES, v -NUMBER, AND CLOSED NEIGHBORHOOD IDEALS 5 The in teger g , denoted by pd ( S/I ), is the pr oje ctive dimension of S/I [ CST + 20 , Vil26 ]. F or the rest of the manuscript, w e iden tify the set of v ariables { t 1 , . . . , t n } with the set of v ertices V ( G ) of a graph G . W e use the notation ⟨ D ⟩ to indicate the ideal generated by the v ariables or vertices in D ⊆ V ( G ). The corresp ondence b etw een v ariables and vertices allo ws us to define families of squarefree monomial ideals parametrized b y subsets of v ertices, suc h as edge ideals [ SVV94 ], which is defined as follows. The e dge ide al of a graph G is denoted and defined b y I ( G ) := ⟨ t i t j | { t i , t j } ∈ E ( G ) ⟩ . W e are interested in the family of closed neighborho o d ideals, whic h naturally extends the notion of edge ideal from a combinatorial persp ective [ SM20 ]. F or a set of vertices D ⊆ V ( G ), the square-free monomial parametrized by D is denoted by t D := Y t j ∈ D t j . The close d neighb orho o d ide al of G , denoted by N G , is defined as N G := ⟨ t N [ t i ] | t i ∈ V ( G ) ⟩ . Remark 2.4. A generator of an edge ideal is the product of a vertex with one of its neigh b ors. A generator of a closed neighborho o d ideal is the pro duct of a v ertex with all of its neighbors. 2.3. Co ding theory. Let F q denote the finite field of order q , where q = p r is a prime p ow er. W e sa y that C is an [ m, k , d ] c o de if C is a linear co de ov er F q of length m , dimension k , and minim um distance d . Observ e that the co de C is a k -dimensional subspace of the vector space F m q and | C | = q k . The code C can correct up to t [ HP10 ], where (2) t :=  d − 1 2  . A Hamming ball S ρ ( x ) of radius ρ with center at the vector x = ( x 1 , . . . , x m ) ∈ F m q is defined as the set S ρ ( x ) := { y ∈ F m q : d ( x, y ) ≤ ρ } , where d ( x, y ) denotes the Hamming distance b etw een the vectors x and y ; this is the num b er of co ordinates in which x and y differ. The w ell-known Hamming b ound can b e stated as follows. Theorem 2.5. [ HP10 ] . L et C b e an [ m, k , d ] c o de and define t as in Equation ( 2 ). Then, | C | t X i =0  m i  ( q − 1) i ≤ q m . If C ac hieves the previous b ound, we say that C is p erfe ct . A p erfect co de can b e understo o d as follo ws. Around each codeword c ∈ C , we draw a Hamming ball S t ( c ) con taining all vectors whose Hamming distance from c is at most t . The elements in the ball represent all p ossible receiv ed co dewords with at most t errors that can b e efficien tly deco ded to c . A code is p erfect if these balls do not ov erlap, so decoding is unam biguous, and the union of the balls cov ers the en tire space F m q . In other w ords, ev ery p ossible receiv ed co deword lies in exactly one Hamming ball, meaning the space is filled p erfectly with no gaps and no o verlaps. 6 JARAMILLO-VELEZ, L ´ OPEZ, AND SAN-JOS ´ E Notation Description G A simple graph N ( v ) Op en neigh b orho o d of the v ertex v N [ v ] Closed neigh b orho o d of the vertex v D c Complemen t of the set of v ertices D P N ( U ) Set of external and self priv ate neigh b ors of U γ ( G ) Domination n umber of the graph G τ ( G ) Co ver num b er of the graph G i ( G ) Indep endence n umber of the graph G a ( G ) Matching num b er of the graph G S P olynomial ring in n v ariables ( I : f ) Colon ideal of I and f Ass( I ) Assosiated primes of I v ( I ) The v -num b er of I reg ( S/I ) Casteln uov o–Mumford regularit y of S/I t A Square-free monomial parametrized b y the v ariables in A ⟨ D ⟩ Ideal generated by the elemen ts in D N G Closed neigh b orho o d ideal associated to G t A | t B The monomial t A divides the monomial t B H q ( r ) Hamming code with redundancy r d ( x, y ) Hamming distance b etw een x , and y Γ( m, q ) Hamming graph representation of F m q T able 1. T able of Notation 3. The v -number of neighborhood ideals In this section, w e present a formula for the v -num b er of a neigh b orho o d ideal. T o this end, w e emplo y the concept of priv ate neighbors, and we lev erage the follo wing description of the minimal primes of the neighborho o d ideal. Prop osition 3.1. [ HSW22 ] . The close d neighb orho o d ide al N G ⊆ S has the fol lowing irr e ducible de c omp ositions: N G = \ D ′ dominating set ⟨ D ′ ⟩ = \ D ′ minimal dominating set ⟨ D ′ ⟩ . F urthermor e, the se c ond interse ction is an irr e dundant de c omp osition. The following relation b etw een the concepts of domination and irredundance is a particularly useful fact for our prop osed form ula of the v -num b er of closed neigh b orho o d ideals. Prop osition 3.2. [ CHM78 ] . If D is minimal dominating, then D is irr e dundant. PRIV A TE NEIGHBORS, PERFECT CODES, v -NUMBER, AND CLOSED NEIGHBORHOOD IDEALS 7 Pr o of. The pro of go es by contradiction. This means w e assume that there is a minimal domi- nating set D that is not an irredundant set. Then, there exists an element d ∈ D such that for all v ∈ V ( G ), d ∈ N [ v ] or N [ v ] \ { d } ⊆ D c . Let’s show that this implies that B := D \ { d } is a dominating set, i.e., for every x ∈ V ( G ), there is y ∈ B that is adjacen t to x . If x = d , then exists y ∈ N [ d ] \ { d } , and y ∈ D c . In this w ay , we ha ve that d is adjacent to an elemen t y in B . No w assume that x  = d and x ∈ D . Since D is a dominating set, exists w ∈ D , with x ∈ N ( w ). If w  = d , then x is adjacent to a v ertex in B . If w = d , then, by the contradiction assumption we obtain that N [ x ] \ { d } ⊆ D c . Th us, there exists y ∈ N [ x ] \ { d } , and y ∈ D . This implies that x is adjacent to y ∈ B , and it sho ws that B is a dominating set. This con tradicts the fact that D is a minimal dominating set. □ Remark 3.3. F or monomial ideals, the v -num b er satisfies an additivity prop ert y analogous to that of Castelnuo vo–Mumford regularity . Sp ecifically , if I 1 ⊂ R 1 and I 2 ⊂ R 2 are monomial ideals in p olynomial rings R 1 and R 2 whose sets of v ariables are disjoin t, then v ( I 1 + I 2 ) = v ( I 1 ) + v ( I 2 ) , where I 1 + I 2 is viewed as an ideal in the p olynomial ring generated by the v ariables of b oth R 1 and R 2 [ SS22 , L TT10 ]. This observ ation shows that, for our analysis of the v-n umber of closed neighborho o d ideals and its connection with Castelnuo vo–Mumford regularity , it suffices to consider connected simple graphs. Remark 3.4. Let A , and D be sets of v ertices, with ( N G : t A ) = ⟨ D ⟩ . W e hav e that for all v ertices d ∈ D , there exists a vertex v suc h that N [ v ] ⊆ A ∪ { d } , and N [ v ] ⊆ A , b ecause ( N G : t A )  = S . This implies that A ∩ D = ∅ . Lemma 3.5. L et N G b e the close d neighb orho o d ide al of a simple c onne cte d gr aph G . The fol lowing hold. (a) F or a minimal dominating set D , and a set C ⊆ P N ( D ) that dominates D , we have that ( N G : t N [ C ] \ D ) = ⟨ D ⟩ . (b) If ( N G : f ) = ⟨ D ⟩ , with D minimal dominating set, and f ∈ S d , then exists a set of vertic es C ⊆ P N ( D ) that dominates D such that ( N G : t N [ C ] \ D ) = ⟨ D ⟩ , and | N [ C ] \ D | ≤ deg ( f ) . Pr o of. (a) W e first start with a monomial t c = t c 1 1 · · · t c n n in ( N G : t N [ C ] \ D ), and let supp ( t c ) denote its supp ort, this is the set of v ariables with p ositive exp onent. This implies that there is a v ertex v suc h that t N [ v ] | t c t N [ C ] \ D . Giv en that D is a dominating set, there exists t j ∈ D with t j ∈ N [ v ] ⊆ supp ( t c t N [ C ] \ D ) = supp ( t c ) ∪ ( N [ C ] \ D ) . Th us, t j ∈ supp ( t c ), which implies that t c ∈ ⟨ D ⟩ . This prov es the containmen t ( N G : t N [ C ] \ D ) ⊆ ⟨ D ⟩ . T o show the rev erse inclusion, let’s consider t j ∈ D . Since C ⊆ P N ( D ) and C dominates D , then there is v j ∈ C such that N [ v j ] ∩ D = { t j } . This implies that N [ v j ] = supp ( t j t N [ v j ] \ D ) ⊆ supp ( t j t N [ C ] \ D ) . Therefore, t j ∈ ( N G : t N [ C ] \ D ), and ⟨ D ⟩ ⊆ ( N G : t N [ C ] \ D ). 8 JARAMILLO-VELEZ, L ´ OPEZ, AND SAN-JOS ´ E (b) Let’s write the p olynomial f as sum of monomials, f = r X i =1 λ i t c i , where 0  = λ i ∈ K and t c i := t c i 1 1 · · · t c in n ∈ S d for all i . Then, w e obtain ( N G : f ) = r \ i =1 ( N G : t c i ) = ⟨ D ⟩ , and consequen tly ( N G : t c k ) = ⟨ D ⟩ for some k . Giv en that N G is a squarefree monomial, w e ma y assume that c kj ∈ { 0 , 1 } for all j , and let’s denote the supp ort of this monomial b y A , this is A := supp ( t c k ) . Therefore, ( N G : t A ) = ⟨ D ⟩ . Moreo ver, we can assume A ∩ D = ∅ b y Remark 3.4 . F or ev ery t j ∈ D , then exists v j ∈ V ( G ) suc h that t N [ v j ] | t j t A . This implies that N [ v j ] \ { t j } ⊆ A . W e consider the set form by al the v j ’s, this is C := { v 1 , . . . , v | D | } . Notice that D ⊆ N [ C ], and | N [ v j ] ∩ D | = 1, b ecause A ∩ D = ∅ . Th us, C ⊆ P N ( D ) and dominates D . Moreov er, ( N G : t N [ C ] \ D ) = ⟨ D ⟩ , b ecause of part (a), and | N [ C ] \ D | ≤ | A | ≤ deg ( f ). □ Theorem 3.6. L et N G b e the close d neighb orho o d ide al of a simple c onne cte d gr aph G . Then, v ( N G ) = min {| N [ C ] \ D | | D is a minimal dominating set, and C ⊆ P N ( D ) dominates D } . Pr o of. W e start by p ointing out that Prop osition 3.1 characterizes the ass o ciated primes of the ideal N G , they are given by the minimal dominating sets. Therefore, to estimate the v -n umber of N G , we focus on a minimal dominating set D of G . F or a pair ( D , C ) such that C ⊆ P N ( D ) and C dominates D , Lemma 3.5 (a) implies that ( N G : t N [ C ] \ D ) = ⟨ D ⟩ . F rom whic h we can conclude that v ( N G ) ≤ min {| N [ C ] \ D | | D is a minimal dominating set, and C ⊆ P N ( D ) dominates D } . T o prov e the reverse inequality assume that v ( N G ) = deg ( f ), with f ∈ S d , and ( N G : f ) = ⟨ D ⟩ . Then, Lemma 3.5 (b) implies that exists a set C ⊆ P N ( D ) and C dominates D with | N [ C ] \ D | ≤ deg ( f ). Thus, v ( N G ) ≥ min {| N [ C ] \ D | | D is a minimial dominating set, and C ⊆ P N ( D ) dominates D } . □ Remark 3.7. F or a fixed minimal dominating set D , to compute min {| N [ C ] \ D | | C ⊆ P N ( D ) dominates D } it is enough to consider subsets C of P N ( D ) with |C | = | D | . Indeed, an y C ⊆ P N ( D ) that dominates D must hav e |C | ≥ | D | , since | N [ c ] ∩ D | = 1 for every c ∈ C . And given C ⊂ C ′ ⊆ P N ( D ), b oth dominating D , then | N [ C ] \ D | ≤ | N [ C ′ ] \ D | . PRIV A TE NEIGHBORS, PERFECT CODES, v -NUMBER, AND CLOSED NEIGHBORHOOD IDEALS 9 Example 3.8. Using Theorem 3.6 , we compute the v -num b er of N K n 1 ,...,n r , the complete r - partite graph, with r ≥ 2 and n 1 ≥ · · · ≥ n r ≥ 2. W e denote its v ertices V n 1 ,...,n r = { v 1 1 , . . . , v 1 n 1 , . . . , v r 1 , . . . , v r n r } . All minimal dominating sets are of the form D k,t i k ,i t = { v k i k , v t i t } , for some 1 ≤ i k ≤ n k , 1 ≤ i t ≤ n t , 1 ≤ k < t ≤ r , or D ( j ) = { v j 1 , . . . , v j n j } , for some 1 ≤ j ≤ r . • W e ha v e P N ( D k,t i k ,i t ) = D ( i k ) ∪ D ( i t ) \ D k,t i k ,i t . Consider C ⊂ P N ( D k,t i k ,i t ) that dominates D k,t i k ,i t . Then it m ust con tain at least one elemen t from D ( i k ) \ { v k i k } (to ensure v t i t ∈ N [ C ]), and also at least one element from D ( i t ) \ { v t i t } (to ensure v k i k ∈ N [ C ]). Then N [ C ] = V n 1 ,...,n r , and | N [ C ] \ D k,t i k ,i t | = P r j =1 n j − 2. • W e obtain P N ( D ( j )) = D ( j ), for 1 ≤ j ≤ r . The only C ⊂ P N ( D ( j )) that dominates D ( j ) is C = D ( j ), and | N [ C ] \ D ( j ) | = P i  = j n i . By Theorem 3.6 , w e get v ( N K n 1 ,...,n r ) = P r i =2 n i . 4. The rela tion between the v -number and other inv ariants This section is dedicated to presenting several relations b etw een the v -num b er and graph in v ariants, as well as to sho wing the relation with the Castelnuo vo-Mumford regularity for sp ecial families of graphs. Theorem 4.1. L et G b e a simple c onne cte d gr aph, and N G its asso ciate d neighb orho o d ide al. Then, γ ( G ) ≤ v ( N G ) ≤ τ ( G ) . Pr o of. F or the first inequality , let ( D, C ) b e a pair with D minimal domination set, C ⊂ P N ( D ) that dominates D , and v ( N G ) = | N [ C ] \ D | . W e write D as follows D = { a 1 , . . . , a r , b 1 , . . . , b s } , where an elements a i has an external priv ate neighbor, and an element b j do es not hav e a external priv ate neigh b or. Then, the set D ′ := [ b j ∈ D N ( b j ) ∪ { a i | a i ∈ D } is a dominating set. Indeed, if v ∈ V ( G ), then v is adjacent to an element in D that has an external priv ate neighbor or v ∈ N ( b j ) for some b j . Notice that N [ C ] \ D con tains a set of cardinalit y | D ′ | , b ecause the graph is connected, then N ( b j ) ⊂ N [ C ] \ D for all b j ∈ D , and N [ C ] \ D con tains an external priv ate neigh b or for every element a i ∈ D . Thus, | N [ C ] \ D | ≥ | D ′ | ≥ γ ( G ) . F or the second inequalit y , let C b e a minimal v ertex co ver with |C | = τ ( G ). The complemen t of D := V ( G ) \ C is a maximal independent set, and therefore a minimal dominating set b y Prop osition 2.2 . Giv en that D is independent we hav e D ⊆ P N ( D ) and D dominates D . Thus, τ ( G ) = |C | = | V ( G ) \ D | = | N [ D ] \ D | ≥ v ( N G ) . □ A particularly interesting case is when γ ( G ) = τ ( G ), since this directly determines v ( N G ). These types of graphs hav e b een studied in the literature [ R V98 , V ol94 ]. In particular, the minim um degree of G has to b e 2, and if it is equal to 2, then G has to b e bipartite [ R V98 ]. W e sho w an example with the complete bipartite graph in Example 4.4 . 10 JARAMILLO-VELEZ, L ´ OPEZ, AND SAN-JOS ´ E Corollary 4.2. L et N G b e the close d neighb orho o d ide al of a simple c onne cte d gr aph G . Then v ( N G ) ≤ 2 a ( G ) . Pr o of. Let M be a matching in G with maximal size, this is a ( G ) = | M | . Notice that the set of vertices in M forms a vertex co ver, otherwise we can obtain a matching of size | M | + 1. Therefore, b y Theorem 4.1 v ( N G ) ≤ γ ( G ) ≤ 2 | M | = 2 a ( G ) . □ Besides the previous b ound, v ( N G ) can b e lo wer than, greater than, or equal to a ( G ), as the next examples show. Moreov er, w e can get the equalit y v ( N G ) = 2 a ( G ) for some graphs. Example 4.3. Consider the graph P 6 giv en b y a path with 6 v ertices { v 1 , . . . , v 6 } . Then, w e ha ve that a ( P 6 ) = 3 and D = { v 2 , v 5 } is a minimal dominating set. If we take C = { v 1 , v 6 } , b y Theorem 3.6 , we hav e that | N [ C ] \ D | = 2. In other words, v ( N P 6 ) ≤ 2 < a ( P 6 ). One can also c heck that, in fact, v ( N P 6 ) = 2. No w consider the complete graph K n , for n ≥ 2. Its matching n umber is a ( K n ) =  n 2  . Let V ( K n ) = { v 1 , . . . , v n } b e the v ertices of K n . Then an y D i = { v i } , 1 ≤ i ≤ n , is a minimal dominating set. The set of priv ate neighbors of D i is V ( K n ). Th us, b y Theorem 3.6 , v ( N K n ) = n − 1. If n ≥ 3, w e obtain v ( N K n ) > a ( K n ). In fact, we obtain v ( N K n ) = ( 2 a ( K n ), if n is o dd, 2 a ( K n ) − 1, if n is ev en. Next w e fo cus in the relation betw een the v -n umber and the Casteln uo vo-Mumford regularit y . Example 4.4. W e consider the complete bipartite graph K n 1 ,n 2 with n 1 ≥ n 2 . F rom Example 3.8 we obtain that v ( N K n 1 ,n 2 ) = n 2 , and from [ SM20 , Thm. 2.10], we hav e reg ( S/ N K n 1 ,n 2 ) = n 1 + n 2 − 2. Therefore, reg ( S/ N K n 1 ,n 2 ) − v ( N K n 1 ,n 2 ) = n 1 − 2 . Moreo ver, a ( K n 1 ,n 2 ) = n 2 , and the difference b et ween the regularit y and the matching num b er can also b e made arbitrarily large. F or edge ideals, it has been sho wn that there exist families of graphs for which the v -num b er is smaller than the Castelnuo v o–Mumford regularit y . On the other hand, there are also fam- ilies of graphs for which the v -n umber exceeds the Castelnuo vo–Mumford regularity , with an arbitrarily large gap b etw een the tw o in v ariants [ Civ23 ]. In the case of closed neighborho o d ideals, Example 4.4 suggests that the Casteln uov o–Mumford regularit y may b e greater than the v -n umber for a broader class of graphs. As we show b elow, this is indeed the case. Moreov er, w e conjecture that this phenomenon holds in general. Theorem 4.5. [ CJRS25 ] . F or any gr aph G , reg ( S/ N I ( G )) ≥ a ( G ) , wher e a ( G ) denotes the matching numb er of G . Mor e over, the e quality holds when G is a tr e e. Theorem 4.6. [ MS26 ] . If G is a bip artite gr aph or a very wel l-c over e d gr aph, then reg ( S/ N G ) ≥ τ ( G ) . Theorem 4.7. If G is a bip artite gr aph or a very wel l-c over e d gr aph, then v ( N G ) ≤ reg ( S/ N G ) . PRIV A TE NEIGHBORS, PERFECT CODES, v -NUMBER, AND CLOSED NEIGHBORHOOD IDEALS 11 Mor e over, if G is a tr e e, v ( N G ) ≤ τ ( G ) = a ( G ) = reg ( S/ N G ) . Pr o of. This result is a consequence of Theorem 4.6 , Theore m 4.1 , and K¨ onig’s theorem [ Die25 , Thm. 2.1.1]. □ Conjecture 4.8. F or any simple c onne cte d gr aph G , we have v ( N G ) ≤ reg ( S/ N G ) . 5. The v -number and perfect codes This section is devoted to presenting a relation b etw ee n the v -n umber of closed neigh b orho o d ideals and the parameters of perfect error-correcting co des. Let’s first present the definition of an efficien t domination set. Definition 5.1. [ HR25 ] . A dominating set D of graph G is called an efficient dominating set if | N [ v ] ∩ D | = 1 for ev ery v ∈ V ( G ) . Efficien t dominating sets, also called p erfect codes, pro vide a connection b etw een graph theory and co ding theory , as the problem of determining the existence of an efficient dominating set in a graph generalizes the problem of determining the existence of a p erfect co de in a finite v ector space, as we explain next [ Big73 ]. The space F m q endo wed with the Hamming distance can b e interpreted as a graph Γ( m, q ) suc h as the set of v ertices corresp onds to the set of p oints of in F m q , this is V (Γ( m, q )) = F m q . And there is an edge betw een t wo vertices of Γ( m, q ) if the corresp onding vectors differ exactly in one entry , this is E (Γ( m, q )) = {{ u, v } | d ( u, v ) = 1 } . This family of graphs is called Hamming graphs, and has b een extensively studied from a sp ectral graph theory p ersp ective [ IKz00 ]. If q = 2, Γ( m, q ) corresp onds to the w ell kno wn h yp ercub e graph [ HHW88 ]. Notices that this construction allows us to establish the classical problem of the existence of p erfect co des as a question in the graph Γ( m, q ). A p erfect co de C correcting t errors is suc h that the Hamming balls S t ( c ) for every codeword of c ∈ C form a partition of the space F m q . If w e let N t ( v ) denote the set of vertices with graph distance t from the v ertex v , then a p erfect co de C is a subset of vertices of Γ( m, q ) such that the sets N t ( c ) ∪ { c } form a partition of the Hamming graph. If t = 1, then the set { N [ c ] | c ∈ C } , forms a partition of Γ( m, q ), this means C is an efficient domination set. In this w ay , an efficien t dominating set corresp onds to a p erfect c o de that corrects 1-errors. The notions of p erfect co des, efficient dominating sets, and priv ate neigh b ors in the graphs ha ve b een an activ e line of researc h considered b y sev eral authors [ Big73 , FFHJ94 , HR25 ]. W e fo cus on the case of p erfect co des that correct 1 error, q -ary Hamming co des. These codes are among the most imp ortant and classical error-correcting co des. In tro duced by Ric hard Hamming, they were the first family of co des to provide a systematic and efficient metho d for detecting and correcting transmission errors. Their significance lies in their practical use in digital comm unication and data storage, and they serve as a cen tral example of p erfect co des, il- lustrating the deep interaction betw een algebra, com binatorics, and information theory [ Ham50 ]. A q -ary Hamming co de, denoted b y H q ( r ), is a linear code with parameters h q r − 1 q − 1 , q r − 1 q − 1 − r, 3 i , 12 JARAMILLO-VELEZ, L ´ OPEZ, AND SAN-JOS ´ E 000 100 110 010 001 101 111 011 Figure 1. The v ertices in red comp ose the Hamming co de H 2 (2). where r ≥ 2 is the num b er of redundant bits in the co de. In graph terms H q ( r ) is an effi- cien t dominating set of the Hamming graph Γ( q r − 1 q − 1 , q ). The following example illustrates the Hamming code in the graph and the construction of Γ( m, q ). Example 5.2. Let’s consider the graph Γ(3 , 2), which corresp onds to the cub e in Figure 1 . In this case, r = 2, which means that we enco de 1 bit b y using tw o m ore redundant bits. The corresp onding Hamming co de in this case is H 2 (2) = { 000 , 111 } . The closed neighborho o ds N [000], and N [111] form a partition of the graph Γ(3 , 2). Next, w e compute the domination num b er, cov er n umber, and indep endence num b er of the Hamming graph Γ( q r − 1 q − 1 , q ), even though these results migh t be well kno wn. W e use these results to bound the v -n umber of Γ( q r − 1 q − 1 , q ) in terms of the redundancy of the Hamming co de H q ( r ). Lemma 5.3. We have that γ  Γ  q r − 1 q − 1 , q  = q q r − 1 q − 1 − r , and γ (Γ ( m, q )) > q m 1 + m ( q − 1) if m  = q r − 1 q − 1 for some r . Pr o of. The code H q ( r ) is a minimal dominating set, and th us γ (Γ( q r − 1 q − 1 , q )) ≤ q q r − 1 q − 1 − r . Assume w e hav e a minimal dominating set D with | D | < q q r − 1 q − 1 − r . Since it is a dominating set, w e hav e | N [ D ] | = q q r − 1 q − 1 . Ho wev er, noting that N [ c ] = q r for any c ∈ Γ( q r − 1 q − 1 , q ) (this is just the size of a Hamming ball with radius 1), w e alw ays hav e the (sphere packing) b ound | N [ D ] | ≤ q r | D | < q q r − 1 q − 1 , a contradiction. The inequality follo ws from the sphere packing b ound for radius 1 (see Theorem 2.5 ) and the non-existence of p erfect codes with distance 3 if m  = q r − 1 q − 1 . □ Lemma 5.4. We have i (Γ ( m, q )) = q m − 1 and τ (Γ ( m, q )) = q m − 1 ( q − 1) . Pr o of. Consider the set L given by the v ectors whose sum of their entries is equal to 0. Then | L | = q m − 1 , and L is a maximal independent set. Moreo ver, it is a maximum indep enden t set. Indeed, w e can split the v ertices of Γ ( m, q ) in to q m − 1 disjoin t sets b y fixing their first m − 1 en tries. The v ertices in each of those sets are connected, since they are at Hamming distance 1 PRIV A TE NEIGHBORS, PERFECT CODES, v -NUMBER, AND CLOSED NEIGHBORHOOD IDEALS 13 0000000 1000000 0001000 0100000 0010000 0000100 0000010 0000001 0001011 0000011 0011011 0101011 1001011 0001001 0001010 0001111 Figure 2. Tw o close neigh b orho o ds of elements in H 2 (3). apart. Therefore, a maximal indep endent set can hav e at most 1 v ertex from each of those sets, that is, its size is b ounded b y q m − 1 . Since L is independent if and only if V \ L is a v ertex cov er, w e ha ve i (Γ ( m, q )) + τ (Γ ( m, q )) = q m , whic h giv es the result. □ Theorem 5.5. L et H q ( r ) b e the Hamming c o de with r e dundancy r . Then, q q r − 1 q − 1 − r ≤ v ( N Γ( q r − 1 q − 1 ,q ) ) ≤ q q r − 1 q − 1 − 1 ( q − 1) . Pr o of. It follo ws from Theorem 4.1 and Lemmas 5.3 and 5.4 . □ Lastly , w e use the next tw o examples to illustrate that the previous upp er bound is sharp. Example 5.6. F ollowing the setting from Example 5.2 , using Theorem 5.5 we get 2 ≤ v ( N Γ(3 , 2) ) ≤ 4 . Using the co de from App endix A , one can chec k that the upp er b ound is sharp in this case, i.e., v ( N Γ(3 , 2) ) = 4. Example 5.7. The well kno wn Hamming co de H 2 (3) is an efficien t dominating set of the graph Γ(7 , 2) [ Ham50 ]. The co de H 2 (3) uses 3 redundant bits to enco de 4 bits, this means it has parameters [7 , 4 , 3]. The Hamming co de can be represented by the follo wing codewords H 2 (3) are H 2 (3) = { 0000000 , 0001011 , 0010111 , 0011100 , 0100110 , 0101101 , 0110001 , 0111010 , 1000101 , 1001110 , 1010010 , 1011001 , 1100011 , 1101000 , 1110100 , 1111111 } . In Figure 2 we display part of the graph Γ(7 , 2) that represen ts t wo close neighborho o ds of t w o of the elements in H 2 (3). Using Theorem 5.5 w e obtain that 16 = 2 4 ≤ v ( N Γ(7 , 2) ) ≤ 2 6 = 64 . References [ASS24] Siddhi Balu Ambhore, Kamalesh Saha, and Indranath Sengupta. The v-n umber of binomial edge ideals. A cta Mathematica Vietnamica , 49(4):611–628, 2024. [Big73] Norman Biggs. Perfect co des in graphs. Journal of Combinatorial The ory, Series B , 15(3):289–296, 1973. [CHM78] Ernest J Co ck ayne, Stephen T Hedetniemi, and Donald J Miller. Prop erties of hereditary hypergraphs and middle graphs. Canadian Mathematic al Bul letin , 21(4):461–468, 1978. 14 JARAMILLO-VELEZ, L ´ OPEZ, AND SAN-JOS ´ E [Civ23] Y usuf Civ an. The v -num b er and Castelnuo vo-Mumford regularity of graphs. J. Algebr aic Combin. , 57(1):161–169, 2023. [CJRS25] Shiny Chakrab orty , Ajay P Joseph, Amit Roy , and Anurag Singh. 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Code f or examples In this section, we provide the Sage co de for the computations that app ear in this pa- p er [ The25 ]. F or the computation of the v -num b er using Theorem 3.6 , w e hav e taken into accoun t Remark 3.7 to make the computation more manageable. W e first provide the basic functions. def closed_neighborhood (G, S): """Returns the closed neighborhood N[S] of a set of vertices S.""" N_S = set (S) for v in S: N_S . update(G . neighbors(v)) return N_S def CNI (G, base_ring = QQ): """Constructs the closed neighborhood ideal of a graph G.""" V = G . vertices(sort = True ) var_names = [ 'x_ {} ' . format(i) for i in range ( len (V))] R = PolynomialRing(base_ring, var_names) gens = R . gens() var_map = {V[i]: gens[i] for i in range ( len (V))} ideal_generators = [] for v in V: c_neighborhood = closed_neighborhood(G, {v}) monomial = R( 1 ) for u in c_neighborhood: monomial *= var_map[u] ideal_generators . append(monomial) return R . ideal(ideal_generators) def v_number (I): """Computes the v-number of a monomial ideal I.""" ass_primes = I . associated_primes() v_num = infinity for P in ass_primes: colon_ideal = I . quotient(P) # Check the minimal generators of the colon ideal 16 JARAMILLO-VELEZ, L ´ OPEZ, AND SAN-JOS ´ E for f in colon_ideal . interreduced_basis(): if I . quotient(f) == P: d = f . degree() if d < v_num: v_num = d return v_num def v_number_formula (G): """ Computes the v-number using the formula from the paper.""" # Precompute all closed neighborhoods V = set (G . vertices()) N_cache = {v: closed_neighborhood(G, {v}) for v in V} min_val = infinity best_D = None best_C = None for D in G . minimal_dominating_sets(): # Group private neighbors by the vertex in D that they belong to PN_dict = {d: [] for d in D} for v in V: intersect = N_cache[v] . intersection(D) if len (intersect) == 1 : # v is a private neighbor of d d = next ( iter (intersect)) # extract the only element in the intersection , → PN_dict[d] . append(v) # By taking one element for each PN(d), we ensure that C dominates D, #and C is contained in PN(d) for C_tuple in itertools . product( * PN_dict . values()): # Compute N[C] N_C = set () for c in C_tuple: N_C . update(N_cache[c]) current_val = len (N_C) - len (D) if current_val < min_val: min_val = current_val best_D = D best_C = set (C_tuple) return min_val, best_D, best_C No w we pro vide the co de to obtain the results from Example 4.4 . n1 = 5 PRIV A TE NEIGHBORS, PERFECT CODES, v -NUMBER, AND CLOSED NEIGHBORHOOD IDEALS 17 n2 = 3 #n1>=m2>=2 from itertools import combinations, chain, product from sage.all import singular singular . lib( 'mregular.lib' ) G = Graph() G = graphs . CompleteBipartiteGraph(n1,n2) # Generate the ideal NI = CNI(G) matching_edges = G . matching() matching_number = len (matching_edges) v_cover = G . vertex_cover() v_cover_size = len (v_cover) vn = v_number(NI) reg = singular . regIdeal(NI) -1 print ( ' \n Matching number: ' , matching_number, 'Minimal vertex cover: ' , v_cover_size, 'v-number: ' , vn, 'Regularity: ' ,reg) , → val, D_min, C_min = v_number_formula(G) print ( 'Formula v-number: ' , val, 'D_min: ' ,D_min, 'C_min: ' , C_min, 'N(C) :' , closed_neighborhood(G, C_min)) , → (Jaramillo-Velez) Dpto. Ma tem ´ aticas, Est ad ´ ıstica e Investigaci ´ on Opera tiv a. Instituto Uni- versit ario de Ma tem ´ aticas y Aplicaciones (IMA ULL). Universid ad de La Laguna. Ap ar t ado de Correos 456. 38200 La La guna, Tenerife, Sp ain, and, Dep ar tment of Ma thema tics, Virginia Tech, Blacksbur g, V A, USA, Email addr ess : djaramil@ull.es, delio@vt.edu (L ´ opez) Dep ar tment of Ma thema tics, Virginia Tech, Blacksburg, V A, USA Email addr ess : hhlopez@vt.edu (San-Jos ´ e) Dep ar tment of Ma thema tics, Virginia Tech, Blacksburg, V A, USA Email addr ess : rsanjose@vt.edu