Graphs with core(G) = nucleus(G)

Graphs with core ( G ) = n ucleus ( G ) V adim E. Levit a , Eugen Mandrescu b , Kevin P ereyra c,d a Dep artment of Mathematics, Ariel University, A riel, Isr ael. b Dep artment of Computer Scienc e, Holon Institute of T e chnolo gy, Holon, Isr ael. c Instituto de Matemátic a Aplic ada San Luis, Universidad Nacional de San Luis and CONICET, San Luis, Ar gentina. d Dep artamento de Matemátic a, Universidad Nacional de San Luis, San Luis, Ar gentina. Abstract Let G b e a finite simple graph. An indep enden t set I of G is critical if | I | − | N ( I ) | ≥ | J | − | N ( J ) | for every indep enden t set J of G . A critical indep enden t set is maxim um if it has maximum cardinalit y . The c or e and the nucleus of G are defined as the intersection of all maxim um indep enden t sets and the in tersection of all maxim um critical indep enden t sets, resp ectiv ely . In 2019, Jarden, Levit and Mandrescu p osed the problem of c haracterizing the graphs satisfying core ( G ) = nucleus ( G ) . In this pap er w e provide a complete solution to this problem. Using Larson’s indep endence decomposition, whic h partitions any graph in to a König–Egerv áry comp onen t L G and a 2 -bicritical comp onen t L c G , w e establish that core ( G ) = n ucleus ( G ) holds if and only if core ( L c G ) = ∅ and no v ertex of corona ( G ) lies in the b oundary b et ween L G and L c G . W e also show that the same b oundary condition is equiv alent to the iden tity diadem ( G ) = corona ( G ) ∩ L ( G ) . Several consequences and related structural prop erties are also derived. Keywor ds: Maxim um Critical Indep enden t Set, Diadem, K önig–Egerv áry graph, Corona, Nucleus, 2-bicritical 2000 MSC: 05C70, 05C75 Email addr esses: levitv@ariel.ac.il (V adim E. Levit), eugen_m@hit.ac.il (Eugen Mandrescu), kdpereyra@unsl.edu.ar (Kevin P ereyra) 1. In tro duction Let α ( G ) denote the cardinalit y of a maximum indep enden t set, and let µ ( G ) b e the size of a maximum matc hing in G = ( V , E ) . It is kno wn that α ( G ) + µ ( G ) equals the order of G , in whic h case G is a K önig–Egerv áry graph [ 3 , 8 , 30 ]. K önig–Egerv áry graphs ha v e b een extensiv ely studied [ 2 , 9 , 11 , 16 , 17 ]. It is known that ev ery bipartite graph is a K önig–Egerv áry graph; this follows from classical results of Kőnig and Egerváry [ 6 , 12 ]. These graphs w ere indep enden tly in tro duced b y Deming [ 3 ], Sterb oul [ 30 ], and Ga vril [ 8 ]. A graph G is considered 2 -bicritical if, after remo ving an y tw o distinct v ertices, the remaining subgraph has a perfect matc hing. The notion of 2 - bicritical graphs was introduced in [ 27 ], and they can b e c haracterized as follo ws. Theorem 1.1 ([ 27 ]) . A gr aph G is 2 -bicritic al if and only if | N ( S ) | > | S | for every nonempty indep endent set S ⊆ V ( G ) . The class of 2 -bicritical graphs can b e regarded as the structural coun ter- part of König–Egerv áry graphs [ 14 , 22 , 23 , 24 ]. It is imp ortan t to note that [ 27 ] sho ws that almost every graph is a 2 -bicritical graph. The main to ol used in this w ork is Larson’s independence decomp osition [ 14 ], whic h partitions a graph into tw o parts: one that induces a K önig– Egerv áry graph L G , and another that induces a 2 -bicritical graph L c G . Let Ω ∗ ( G ) = { S : S is an indep enden t set of G } , Ω( G ) = { S : S is a maxim um indep enden t set of G } , core ( G ) = T { S : S ∈ Ω( G ) } [ 15 ], and corona ( G ) = S { S : S ∈ Ω( G ) } [ 1 ]. The num b er d G ( X ) = | X | − | N ( X ) | is the difference of the set X ⊂ V ( G ) , and d ( G ) = max { d G ( X ) : X ⊂ V ( G ) } is called the critic al differ enc e of G . A set U ⊂ V ( G ) is critic al if d G ( U ) = d ( G ) [ 31 ]. The n umber d I ( G ) = max { d G ( X ) : X ∈ Ω ∗ ( G ) } is called the critic al indep endenc e differ enc e of G . If a set X ⊂ Ω ∗ ( G ) satisfies d G ( X ) = d I ( G ) , then it is called a critic al indep endent set [ 31 ]. Clearly , d ( G ) ≥ d I ( G ) holds for every graph. It is known that d ( G ) = d I ( G ) for all graphs [ 31 ]. W e define CritIndep ( G ) = { S : S is a critical indep enden t set of G } and MaxCritIndep ( G ) = { S : S is a maxim um critical indep enden t set of G } . Recall the following: k er ( G ) = T CritIndep ( G ) [ 18 , 21 , 28 ], nucleus ( G ) = T MaxCritIndep ( G ) [ 10 ], and diadem ( G ) = S CritIndep ( G ) [ 29 ]. A ctually , ev ery critical indep enden t set is contained in a maxim um critical independent set, and a maxim um critical independent set can b e found in polynomial time [ 13 ]. Also note that diadem ( G ) = S CritIndep ( G ) = S MaxCritIndep ( G ) . 2 Sev eral in teresting phenomena o ccur when core ( G ) is a critical indep en- den t set, and these hav e b een studied in [ 10 ]. In [ 20 ], the graphs for which core ( G ) is a critical indep enden t set are completely characterized. Theorem 1.2 ([ 20 ]) . F or every gr aph, core ( G ) is a critic al indep endent set if and only if core ( L c G ) = ∅ . Ho wev er, the condition in Theorem 1.2 do es not ensure the equalit y core ( G ) = nucleus ( G ) , as demonstrated by the example in Fig. 1 . Figure 1: A graph satisfying core ( L c G ) = ∅ while core ( G )  = nucleus ( G ) . When core ( G ) is a critical indep enden t set, it is kno wn that core ( G ) ⊆ n ucleus ( G ) [ 10 ], and equality is v alid when diadem ( G ) = corona ( G ) . Conse- quen tly , in [ 10 ] the following problem is p osed: Problem 1.3 ([ 10 ]) . Char acterize the gr aphs enjoying core ( G ) = n ucleus ( G ) . It is known that every König–Egerv áry graph satisfies core ( G ) = n ucleus ( G ) [ 17 ]. In this w ork we solv e Theorem 1.3 b y sho wing that, in addition to the condition core ( L c G ) = ∅ from Theorem 1.2 , the only extra obstruction is the presence of v ertices of corona ( G ) on the b oundary b et w een the tw o sides of Larson’s partition. Our app ro ach also incorp orates the diadem: w e prov e that corona ( G ) ∩ ∂ L ( G ) = ∅ is equiv alen t to diadem ( G ) = corona ( G ) ∩ L ( G ) . The pap er is organized as follo ws . In Section 1 w e present the general con text of the problem and in tro duce the main concepts and definitions. In Section 2 we fix the notation that will b e used throughout the pap er. In Section 3 w e establish the characterization of graphs with core ( G ) = n ucleus ( G ) , together with companion results describing the corresp onding diadem, thereb y solving Theorem 1.3 . Finally , Section 5 discusses concluding remarks and directions for further researc h. 3 2. Preliminaries All graphs considered in this pap er are finite, undirected, and simple. F or any undefined terminology or notation, we refer the reader to Lov ász and Plummer [ 26 ] or Diestel [ 4 ]. Let G = ( V , E ) b e a simple graph, where V = V ( G ) is the finite set of v ertices and E = E ( G ) is the set of edges. A subgraph of G is a graph H suc h that V ( H ) ⊆ V ( G ) and E ( H ) ⊆ E ( G ) . A subgraph H of G is called a sp anning subgraph if V ( H ) = V ( G ) . F or tw o v sets X , Y ⊆ V ( G ) , we denote b y E ( X, Y ) the set of edges uv ∈ E ( G ) suc h that u ∈ X and v ∈ Y . Let e ∈ E ( G ) and v ∈ V ( G ) . W e define G − e = ( V , E − { e } ) and G − v = ( V − { v } , { uw ∈ E : u, w  = v } ) . If X ⊆ V ( G ) , the induc e d subgraph of G b y X is the subgraph G [ X ] = ( X, F ) , where F = { uv ∈ E ( G ) : u, v ∈ X } . The union of tw o graphs G and H is the graph G ∪ H with V ( G ∪ H ) = V ( G ) ∪ V ( H ) and E ( G ∪ H ) = E ( G ) ∪ E ( H ) . The num b er of v ertices in a graph G is called the or der of the graph and is denoted n ( G ) . A cycle in G is called o dd (resp. even ) if it has an o dd (resp. even) num b er of edges. F or a v ertex v ∈ V ( G ) , the neighb orho o d of v is N G ( v ) = { u ∈ V ( G ) : uv ∈ E ( G ) } . When no confusion arises, we write N ( v ) instead of N G ( v ) . F or a set S ⊆ V ( G ) , the neighb orho o d of S is N G ( S ) = [ v ∈ S N G ( v ) . A matching M in a graph G is a set of pairwise non-adjacent edges. The matching numb er of G , denoted b y µ ( G ) , is the maxim um cardinalit y of an y matching in G . Matchings induce an in volution on the v ertex set of the graph: M : V ( G ) → V ( G ) , where M ( v ) = u if uv ∈ M , and M ( v ) = v otherwise. If S , U ⊆ V ( G ) with S ∩ U = ∅ , we sa y that M is a matc hing from S to U if M ( S ) ⊆ U . A matching M is p erfe ct if M ( v )  = v for every v ertex of the graph. A matching is ne ar-p erfe ct if | v ∈ V ( G ) : M ( v ) = v | = 1 . A graph is a factor-critical graph if G − v has p erfect matc hing for ev ery v ertex v ∈ V ( G ) . The deficiency of a graph G , denoted b y def ( G ) , is defined as def ( G ) = | G | − 2 µ ( G ) . 4 A vertex set S ⊆ V is indep endent if, for ev ery pair of v ertices u, v ∈ S , w e hav e uv / ∈ E . The n umber of vertices in a maxim um indep endent set is denoted b y α ( G ) . The Gallai–Edmonds decomposition will play an imp ortant role in this w ork. Theorem 2.1 ([ 5 , 7 ] Gallai–Edmonds structure theorem) . L et G b e a gr aph, and define D ( G ) = { v : there exists a maximum matching that misses v } , A ( G ) = { v : v is adjacen t to some u ∈ D ( G ) , but v / ∈ D ( G ) } , C ( G ) = V ( G ) − ( D ( G ) ∪ A ( G )) . If G 1 , . . . , G k ar e the c onne cte d c omp onents of G [ D ( G )] and M is a maximum matching of G , then: 1. M c overs C ( G ) and matches A ( G ) into distinct c omp onents of G [ D ( G )] . 2. Each G i is a factor-critic al gr aph, and the r estriction of M to G i is a ne ar-p erfe ct matching. 3. Each nonempty S ⊆ A ( G ) is adjac ent to at le ast | S | + 1 c omp onents of G [ D ( G )] . 3. Main Results In this section w e establish the main result of the pap er. W e first pro v e sev eral preliminary lemmas, and then obtain a c haracterization of the graphs whose core equals the nucleus. In [ 14 ], Larson in tro duces the follo wing de- comp osition theorem. Theorem 3.1 ([ 14 ]) . F or any gr aph G , ther e is a unique set L ( G ) ⊂ V ( G ) such that 1. α ( G ) = α ( G [ L ]) + α ( G [ V ( G ) − L ( G )]) , 2. G [ L ( G )] is a König-Egerváry gr aph, 3. for every non-empty indep endent set I in G [ V ( G ) − L ( G )] , we have | N ( I ) | > | I | , and 4. for every maximum critic al indep endent set J of G , L ( G ) = J ∪ N ( J ) . 5 Throughout the remainder of the pap er, L ( G ) and L c ( G ) = V ( G ) − L ( G ) denote the sets of Theorem 3.1 ; moreo ver, to simplify the notation, we define the induced graphs L G = G [ L ( G )] , L c G = G [ L c ( G )] . Observ ation 3.2. By The or em 3.1 , for every gr aph G with L c ( G )  = ∅ , it fol lows that L c G is a 2 -bicritic al gr aph. Lemma 3.3 ([ 13 ]) . L et I b e a critic al indep endent set of G . Then ther e exists a maximum matching of G that matches N ( I ) into I . W e no w recall a structural description of core ( G ) in terms of Larson’s partition. This result allo ws us to separate the con tribution of the tw o sides L G and L c G . Lemma 3.4 ([ 20 ]) . F or every gr aph core ( G ) = core ( L c G ) ∪ ( core ( G ) ∩ L ( G )) . Mor e over, core ( G ) ∩ L ( G ) = \ S ∈ Ω( G ) ( S ∩ L ( G )) ⊆ n ucleus ( G ) . Theorem 3.5 ([ 25 ]) . The e quality k er ( G ) = D ( L G ) holds for every gr aph G . A t this p oin t, the main difficulty b ecomes apparen t. While core ( G ) and MaxCritIndep ( G ) are largely con trolled by the structure of L G , vertices that lie on the in terface b etw een L G and L c G ma y interfere with this behavior. T o capture this phenomenon, w e introduce the b oundary set ∂ L ( G ) = { v ∈ L ( G ) : N ( v ) ∩ L c ( G )  = ∅} . By Theorem 3.1 , it follo ws that ∂ L ( G ) ⊆ N ( I ) for ev ery I ∈ MaxCritIndep ( G ) . Lemma 3.6. If I is a maximum critic al indep endent set of G , then core ( G ) ∩ L ( G ) = M ( N ( I ) − corona ( G )) ∪ D ( L G ) , for every maximum matching M of L G . 6 Pr o of. Let I ∈ MaxCritIndep ( G ) and let M b e a maximum matc hing of L G . By Theorem 3.1 , L ( G ) = I ∪ N ( I ) and L G is a K önig–Egerv áry graph. By Theorem 3.3 , there exists a maximum matc hing of G that matches N ( I ) in to I . Its restriction to L G is a matc hing of cardinalit y | N ( I ) | , and hence µ ( L G ) ≥ | N ( I ) | . Since I is an indep enden t set of L G and α ( L G ) + µ ( L G ) = | L ( G ) | = | I | + | N ( I ) | , w e obtain | I | ≤ α ( L G ) = | L ( G ) | − µ ( L G ) ≤ | I | + | N ( I ) | − | N ( I ) | = | I | . Therefore, I ∈ Ω( L G ) and µ ( L G ) = | N ( I ) | . Consequen tly , ev ery maximum matc hing of L G has cardinality | N ( I ) | . Since I is indep enden t, no edge of M lies inside I . If some edge of M had b oth endp oints in N ( I ) , then M w ould co ver at least | N ( I ) | + 1 v ertices of N ( I ) , whic h is imp ossible. Th us, every edge of M joins a v ertex of N ( I ) to a v ertex of I , and M matches N ( I ) in to I . Let v ∈ core ( G ) ∩ L ( G ) . By Theorem 3.4 , w e ha ve v ∈ n ucleus ( G ) , and hence v ∈ I . If v / ∈ D ( L G ) , then ev ery maximum matching of L G co vers v . In particular, there exists u ∈ N ( I ) such that M ( u ) = v . If u ∈ corona ( G ) , then some set S ∈ Ω( G ) con tains u . Since v ∈ core ( G ) , the same set S also con tains v , con tradicting the fact that uv ∈ E ( G ) . Therefore, u / ∈ corona ( G ) , and so v ∈ M ( N ( I ) − corona ( G )) . This pro ves that core ( G ) ∩ L ( G ) ⊆ M ( N ( I ) − corona ( G )) ∪ D ( L G ) . No w let S ∈ Ω( G ) . By Theorem 3.1 , the set S ∩ L ( G ) b elongs to Ω( L G ) . Since M has | N ( I ) | edges and its unmatc hed v ertices lie in I , every set in Ω( L G ) con tains exactly one endp oin t of eac h edge of M and ev ery unmatc hed v ertex of M . Therefore, if x ∈ D ( L G ) , then there exists a maxim um matching M x of L G that misses x , and the previous observ ation applied to M x sho ws that x ∈ S ∩ L ( G ) . As S ∈ Ω( G ) was arbitrary , it follo ws that D ( L G ) ⊆ core ( G ) ∩ L ( G ) . Finally , let v ∈ M ( N ( I ) − corona ( G )) , and choose u ∈ N ( I ) − corona ( G ) suc h that M ( u ) = v . F or every set S ∈ Ω( G ) , the set S ∩ L ( G ) lies in Ω( L G ) and therefore contains exactly one endp oint of eac h edge of M . Since u / ∈ corona ( G ) , no maxim um indep endent set of G con tains u , and hence ev ery suc h set S m ust con tain v . Therefore, v ∈ core ( G ) ∩ L ( G ) , and this pro ves the reverse inclusion. As a consequence of Theorem 3.4 , Lemma 3.6 , and Theorem 3.5 , w e infer the follo wing. 7 Theorem 3.7. If I is a maximum critic al indep endent set of G , then core ( G ) = core ( L c G ) ∪ M ( N ( I ) − corona ( G )) ∪ k er ( G ) , for every maximum matching M of L G . Theorem 3.8 ([ 17 ]) . G is a König–Egerváry gr aph if and only if e ach of its maximum indep endent sets is critic al. F rom Theorem 3.8 the follo wing is obtained directly . Theorem 3.9. If G is a König–Egerváry gr aph, then MaxCritIndep ( G ) = Ω( G ) , that is, nucleus ( G ) = core ( G ) . Theorem 3.10 ([ 20 ]) . The e quality d ( G ) = d ( L G ) holds for every gr aph G . Ha ving describ ed core ( G ) , w e now turn to the family of maximum critical indep enden t sets. Theorem 3.11. The e quality MaxCritIndep ( G ) = { S ∈ Ω ( L G ) : E ( S, L c ( G )) = ∅} is valid for every gr aph G . Pr o of. Let I ∈ MaxCritIndep ( G ) . By Theorem 3.1 , w e hav e L ( G ) = I ∪ N ( I ) and L G is a K önig–Egerv áry graph. By Theorem 3.3 , there exists a maximum matc hing of G that matc hes N ( I ) in to I . As in the proof of Theorem 3.6 , this implies that µ ( L G ) ≥ | N ( I ) | . Since I is indep endent in L G and α ( L G ) + µ ( L G ) = | L ( G ) | = | I | + | N ( I ) | , we conclude that I ∈ Ω( L G ) . Moreo ver, note that E ( I , L c ( G )) = ∅ . Con versely , let I ∈ Ω ( L G ) such that E ( I , L c ( G )) = ∅ . By Theorem 3.9 w e ha v e I ∈ MaxCritIndep ( L G ) , and therefore | I | − | N L G ( I ) | = d ( L G ) . Note that N L G ( I ) = N G ( I ) , since E ( I , L c ( G )) = ∅ . In addition, by Theorem 3.10 , w e ha v e d ( G ) = d ( L G ) . Therefore, | I | − | N G ( I ) | = d ( G ) , and hence I is a critical indep endent set of G . If J ∈ MaxCritIndep ( G ) , then the first part of the pro of sho ws that J ∈ Ω( L G ) . Thus | J | = α ( L G ) = | I | , so I is a maxim um critical indep enden t set of G . 8 Corollary 3.12. F or every gr aph G , n ucleus ( G ) = \ { S ∈ Ω ( L G ) : E ( S, L c ( G )) = ∅} and diadem ( G ) = [ { S ∈ Ω ( L G ) : E ( S, L c ( G )) = ∅} ⊆ corona ( G ) ∩ L ( G ) . Pr o of. The first equalit y is an immediate reformulation of Theorem 3.11 . The second follows from the definition of diadem ( G ) and the same c haracteriza- tion of MaxCritIndep ( G ) . If I ∈ MaxCritIndep ( G ) , then by Theorem 3.11 w e ha v e I ∈ Ω( L G ) and E ( I , L c ( G )) = ∅ . Let R ∈ Ω( L c G ) . Since I ∪ R is indep enden t and, by Theorem 3.1 , | ( | I ∪ R ) = α ( L G ) + α ( L c G ) = α ( G ) , it follo ws that I ∪ R ∈ Ω( G ) . Hence I ⊆ corona ( G ) ∩ L ( G ) , and taking unions o ver all sets I ∈ MaxCritIndep ( G ) yields the desired inclusion. Theorem 3.13. If I is a maximum critic al indep endent set of G , then M ( N ( I ) − ( corona ( G ) − ∂ L ( G ))) ∪ k er ( G ) ⊆ nucleus ( G ) holds for every maximum matching M of L G . Pr o of. Let I ∈ MaxCritIndep ( G ) . By Theorem 3.1 , we kno w that L ( G ) = I ∪ N ( I ) . Let M b e a maxim um matching of L G . As in the pro of of Theorem 3.6 , M matc hes N ( I ) in to I , and every maxim um indep enden t set of L G con tains exactly one endp oin t of each edge of M . By Theorem 3.11 , n ucleus ( G ) = \ { S ∈ Ω ( L G ) : E ( S, L c ( G )) = ∅} . Let v ∈ M ( N ( I ) − ( corona ( G ) − ∂ L ( G ))) . Cho ose u ∈ N ( I ) − ( corona ( G ) − ∂ L ( G )) suc h that M ( u ) = v . If u / ∈ corona ( G ) , then Theorem 3.6 yields v ∈ core ( G ) ∩ L ( G ) ⊆ n ucleus ( G ) . Supp ose no w that u ∈ ∂ L ( G ) . Let S ∈ Ω ( L G ) satisfy E ( S, L c ( G )) = ∅ . Since u has a neighbor in L c ( G ) , w e m ust ha ve u / ∈ S . Because S con tains exactly one endpoint of each edge of M , it follo ws that v = M ( u ) ∈ S . As S w as arbitrary , we conclude that v ∈ nucleus ( G ) . Finally , since k er ( G ) ⊆ n ucleus ( G ) , we obtain M ( N ( I ) − ( corona ( G ) − ∂ L ( G ))) ∪ k er ( G ) ⊆ nucleus ( G ) , whic h completes the pro of. 9 Theorem 3.14. If G is a König–Egerváry gr aph, then n ucleus ( G ) = M ( N ( I ) − corona ( G )) ∪ ker ( G ) . Pr o of. Let I ∈ MaxCritIndep ( G ) . Since G is a K önig–Egerv áry graph, Theo- rem 3.9 implies that I ∈ Ω( G ) . Hence I ∪ N ( I ) = V ( G ) , and by Theorem 3.1 w e obtain L ( G ) = V ( G ) , that is, L G = G . Again b y Theorem 3.9 , w e ha ve n ucleus ( G ) = core ( G ) . Therefore, Theorems 3.5 and 3.6 yield n ucleus ( G ) = core ( G ) = M ( N ( I ) − corona ( G )) ∪ D ( L G ) = M ( N ( I ) − corona ( G )) ∪ ker ( G ) . W e no w arriv e at the k ey p oint of the argumen t. The follo wing theo- rem shows that, once the b oundary obstruction disapp ears, b oth the family of maximum critical indep endent sets and the corresp onding in v arian ts are completely determined b y the König–Egerv áry part L G . Theorem 3.15. L et G b e a gr aph such that corona ( G ) ∩ ∂ L ( G ) = ∅ . Then MaxCritIndep ( G ) = { T ∩ L ( G ) : T ∈ Ω( G ) } . Conse quently, n ucleus ( G ) = core ( G ) ∩ L ( G ) and diadem ( G ) = corona ( G ) ∩ L ( G ) . Pr o of. By Theorem 3.11 , MaxCritIndep ( G ) = { S ∈ Ω ( L G ) : E ( S, L c ( G )) = ∅} . W e claim that { S ∈ Ω ( L G ) : E ( S, L c ( G )) = ∅} = { T ∩ L ( G ) : T ∈ Ω( G ) } . Let S ∈ Ω ( L G ) satisfy E ( S, L c ( G )) = ∅ . If R ∈ Ω ( L c G ) , then S ∪ R is an indep enden t set of cardinalit y α ( L G ) + α ( L c G ) = α ( G ) , and hence S ∪ R ∈ Ω( G ) . Thus S = ( S ∪ R ) ∩ L ( G ) . Conv ersely , let T ∈ Ω( G ) . By Theorem 3.1 , the set T ∩ L ( G ) b elongs to Ω( L G ) . Moreov er, T ∩ L ( G ) ⊆ corona ( G ) ∩ L ( G ) . Since corona ( G ) ∩ ∂ L ( G ) = ∅ , no vertex of T ∩ L ( G ) is adjacen t to L c ( G ) , 10 and therefore E ( T ∩ L ( G ) , L c ( G )) = ∅ . This pro ves the claim. T aking in tersections and unions ov er the tw o equal families yields n ucleus ( G ) = core ( G ) ∩ L ( G ) and diadem ( G ) = corona ( G ) ∩ L ( G ) , as required. Com bining the previous results, we obtain the main characterization the- orem. Theorem 3.16. A gr aph G satisfies core ( G ) = n ucleus ( G ) if and only if core ( L c G ) = corona ( G ) ∩ ∂ L ( G ) = ∅ . Pr o of. Let I ∈ MaxCritIndep ( G ) . By Theorem 3.1 , L ( G ) = I ∪ N ( I ) . Since n ucleus ( G ) ⊆ I ⊆ L ( G ) and core ( G ) = nucleus ( G ) , it follo ws that core ( G ) ⊆ L ( G ) . By Theorem 3.4 , core ( G ) = core ( L c G ) ∪ ( core ( G ) ∩ L ( G )) , and therefore core ( L c G ) = ∅ . Supp ose no w that there exists u ∈ corona ( G ) ∩ ∂ L ( G ) . Let M be a maxim um matching of L G . As in the pro of of Th eorem 3.6 , the matching M matc hes N ( I ) into I . Set v = M ( u ) . Since u ∈ corona ( G ) , some maximum indep enden t set T of G con tains u . Because uv ∈ E ( G ) , we ha ve v / ∈ T , and hence v / ∈ core ( G ) = nucleus ( G ) . On the other hand, let J ∈ MaxCritIndep ( G ) . By Theorem 3.11 , w e hav e J ∈ Ω ( L G ) and E ( J, L c ( G )) = ∅ . Since u ∈ ∂ L ( G ) , it follo ws that u / ∈ J . As in the pro of of Theorem 3.6 , ev ery maxim um indep endent set of L G con tains exactly one endp oint of each edge of M . Therefore, v = M ( u ) ∈ J . Since J w as arbitrary , v ∈ nucleus ( G ) , a con tradiction. Hence corona ( G ) ∩ ∂ L ( G ) = ∅ . Con versely , supp ose that core ( L c G ) = corona ( G ) ∩ ∂ L ( G ) = ∅ . Then The- orem 3.4 implies that core ( G ) = core ( G ) ∩ L ( G ) . Hence, b y Theorem 3.15 , core ( G ) = core ( G ) ∩ L ( G ) = nucleus ( G ) , as claimed. Corollary 3.17. A gr aph G satisfies diadem ( G ) = corona ( G ) ∩ L ( G ) if and only if corona ( G ) ∩ ∂ L ( G ) = ∅ . 11 Pr o of. If corona ( G ) ∩ ∂ L ( G ) = ∅ , then Theorem 3.15 yields diadem ( G ) = corona ( G ) ∩ L ( G ) . Con v ersely , assume that diadem ( G ) = corona ( G ) ∩ L ( G ) and supp ose that there exists u ∈ corona ( G ) ∩ ∂ L ( G ) . Since u ∈ L ( G ) , the assumed equalit y implies that u ∈ diadem ( G ) , and therefore some set J ∈ MaxCritIndep ( G ) contains u . How ever, Theorem 3.11 yields E ( J , L c ( G )) = ∅ , which is imp ossible b ecause u ∈ ∂ L ( G ) . Hence corona ( G ) ∩ ∂ L ( G ) = ∅ . Corollary 3.18. A gr aph G satisfies core ( G ) = n ucleus ( G ) if and only if core ( L c G ) = ∅ and diadem ( G ) = corona ( G ) ∩ L ( G ) . Pr o of. This follo ws immediately from Theorems 3.16 and 3.17 . Theorem 3.16 completes the solution of Theorem 1.3 . It reveals that the b eha vior of core ( G ) , n ucleus ( G ) , and diadem ( G ) is completely gov erned b y Larson’s decomp osition. An almost-bipartite graph is a graph con taining a unique o dd cycle. In an almost-bipartite non-König–Egerv áry graph, the Larson decomposition is known: L G is a bipartite graph, while L c G is an o dd cycle [ 20 ]. Then core ( L c G ) = ∅ . A dditionally , L G can b e decomp osed via the Gallai-Edmonds structure: D ( L G ) , A ( L G ) , C ( L G ) . It is easy to see by Theorem 2.1 that D ( L G ) ∩ ∂ L ( G ) = A ( L G ) ∩ corona ( G ) = ∅ . Therefore, b y Theorem 3.16 the following holds. Theorem 3.19. A n almost bip artite non-König–Egerváry gr aph G satisfies core ( G ) = nucleus ( G ) if and only if C ( L G ) ∩ corona ( G ) ∩ ∂ L ( G ) = ∅ . 4. Conclusions In this paper w e solv ed the problem posed in [ 10 ] of characterizing the graphs satisfying core ( G ) = n ucleus ( G ) . The solution is expressed in terms of Larson’s indep endence decomp osition V ( G ) = L ( G ) ∪ L c ( G ) . Our main theorem sho ws that the equality core ( G ) = nucleus ( G ) is go v erned b y t wo indep enden t obstructions: the presence of v ertices in core ( L c G ) , whic h 12 comes from the 2 -bicritical side, and the presence of v ertices of corona ( G ) on the b oundary ∂ L ( G ) , whic h measures the in teraction b et ween the t w o parts. The auxiliary results obtained along the w ay pro vide a more detailed description of the maxim um critical structure of a graph. In particular, Theorem 3.11 iden tifies the family MaxCritIndep ( G ) with the maxim um in- dep enden t sets of L G that a void L c G , while Theorems 3.15 and 3.17 sho w that the b oundary condition corona ( G ) ∩ ∂ L ( G ) = ∅ is exactly the condition un- der whic h the maxim um critical sets are the traces on L ( G ) of the maxim um indep enden t sets of G . Consequen tly , n ucleus ( G ) = core ( G ) ∩ L ( G ) and diadem ( G ) = corona ( G ) ∩ L ( G ) whenev er no v ertex of corona ( G ) lies on the Larson b oundary . F or K önig– Egerv áry graphs this recov ers the classical identities nucleus ( G ) = core ( G ) and diadem ( G ) = corona ( G ) . It is worth noting that diadem ( G ) = corona ( G ) if and only if G is a K önig–Egerv áry graph [ 19 , 29 ]. The almost-bipartite case illustrates the usefulness of the characteriza- tion. Since L c G is an o dd cycle in that setting, the obstruction coming from core ( L c G ) disapp ears automatically , and the problem reduces to a condition on the Gallai–Edmonds decomp osition of the K önig–Egerv áry part L G . This suggests that Larson’s decomposition provides a robust framework for fur- ther inv estigations relating maxim um indep endent sets and maximum critical indep enden t sets. 5. Op en problems The results of this pap er suggest several natural directions for further researc h. Problem 5.1. Give an explicit structur al description of diadem ( G ) for an arbitr ary gr aph G in terms of the L arson de c omp osition V ( G ) = L ( G ) ∪ L c ( G ) and the b oundary set ∂ L ( G ) . Problem 5.2. Determine the algorithmic c omplexity of de ciding whether a gr aph G satisfies core ( G ) = nucleus ( G ) . Mor e gener al ly, study the c omplexity of c omputing n ucleus ( G ) and diadem ( G ) fr om a L arson de c omp osition. Problem 5.3. Develop weighte d analo gues of the notions of c or e, nucleus, c or ona, and diadem, and char acterize when the weighte d c or e c oincides with the weighte d nucleus. 13 A c knowledgmen ts This work was partially sup p orted b y Universidad Nacional de San Luis, gran ts PROICO 03-0723 and PROIPR O 03-2923, MA TH AmSud, gran t 22- MA TH-02, Consejo Nacional de In v estigaciones Cien tíficas y Técnicas gran t PIP 11220220100068CO and Agencia I+D+I grants PICT 2020-00549 and PICT 2020-04064. Declaration of generativ e AI and AI-assisted tec hnologies in the writing pro cess During the preparation of this work the authors used ChatGPT-3.5 in order to improv e the grammar of several paragraphs of the text. After using this service, the authors review ed and edited the conten t as needed and tak e full resp onsibilit y for the conten t of the publication. Data av ailabilit y Data sharing not applicable to this article as no datasets w ere generated or analyzed during the curren t study . Declarations Conflict of interest The authors declare that they ha ve no conflict of in terest. References [1] Boros, E., Golum bic, M. C., and Levit, V. E. (2002). On the num b er of v ertices b elonging to all maximum stable sets of a graph. Discr ete Applie d Mathematics , 124(1-3):17–25. 2 [2] Bourjolly , J.-M., Hammer, P . L., and Simeone, B. (2009). No de-weigh ted graphs ha ving the König–Egerváry prop ert y . In Mathematic al Pr o gr am- ming at Ob erwolfach II , pages 44–63. 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