Packing Independent Cliques in $K_4$-minor-free Graphs

Let $G$ be a graph and $S$ be a set of cliques of $G$. The set $S$ is an indeque set if every component of $G[S]$, the subgraph induced by vertices of $S$, is a clique. In this paper, we prove that the indeque ratio of $K_4$-minor-free graphs is $\frac 1 2$, which settle two conjectures of Biro, Collado and Zamora. We also show that the indeque ratio of subcubic graphs is $\frac 1 2$.

💡 Research Summary

The paper studies the maximum size of an “indeque set” – a collection of cliques whose induced subgraph consists of disjoint cliques – in two important families of sparse graphs: K₄‑minor‑free graphs and subcubic graphs (graphs of maximum degree three). The authors denote by αω(G) the largest cardinality of an indeque set in a graph G, and define the indeque ratio of a graph class 𝔊 as the limit inferior of αω(G)/|V(G)| over all n‑vertex members of 𝔊.

Previous work by Biro, Collado and Zamora established lower bounds for planar graphs (αω(G) ≥ 4n/15) and conjectured that the indeque ratio for series‑parallel graphs and for outerplanar graphs should be exactly ½. The present paper resolves both conjectures and, more generally, proves that the indeque ratio of the whole class of K₄‑minor‑free graphs is ½. It also shows that subcubic graphs enjoy the same ratio.

The core of the proof relies on a structural decomposition of K₄‑minor‑free graphs. Such graphs are exactly the graphs of tree‑width at most two; consequently every block (maximal 2‑connected subgraph) is a series‑parallel graph. The authors introduce a hierarchy of “k‑series pieces” and “k‑parallel pieces”, built recursively from elementary 0‑series pieces (paths whose internal vertices have degree two) and 0‑parallel pieces (small 2‑connected subgraphs). They also define a “contrapair” (X, S): a subgraph X together with an indeque set S⊆V(X) satisfying |S| ≥ |X|/2 and such that no vertex of S has a neighbor outside X. If a graph G contains a contrapair, then removing X leaves a smaller K₄‑minor‑free graph G−X; by minimality of a counterexample, G−X has an indeque set of size at least half its vertices, and together with S this yields an indeque set of size at least |V(G)|/2, contradicting the assumption that G is a counterexample.

The authors assume a minimal counterexample G and focus on a leaf block H with a cut‑vertex v. Lemma 2.5 shows that H cannot contain adjacent degree‑2 vertices (otherwise a trivial contrapair exists) and that every 0‑parallel piece of H must be one of six small configurations (illustrated in the paper). Using these restrictions, they analyze all possible shapes of H: triangle‑strings, triangle‑rings, and “kites” (a path attached to a triangle‑string). For each shape they explicitly construct a suitable indeque set S that satisfies the contrapair condition. The analysis proceeds by exhaustive case work, showing that any leaf block inevitably yields a contrapair, which eliminates the possibility of a minimal counterexample. Hence every K₄‑minor‑free graph G satisfies αω(G) ≥ |V(G)|/2, establishing the indeque ratio ½ for the class.

The paper then extends the argument to subcubic graphs. Because the maximum degree is three, any block in a subcubic graph is small enough that the same decomposition and contrapair construction apply. Consequently, the same lower bound αω(G) ≥ |V(G)|/2 holds, and the indeque ratio of subcubic graphs is also ½.

The results settle the two conjectures of Biro, Collado and Zamora, confirming that series‑parallel and outerplanar graphs both have indeque ratio ½, and they provide a unified proof for the broader class of K₄‑minor‑free graphs. The paper contributes a clean structural technique—combining tree‑width based decomposition with the contrapair method—that may be useful for other extremal problems on sparse graph families.