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

TL;DR

The paper resolves the indeque ratio for the class of $K_4$-minor-free graphs by proving a tight lower bound $^{\alpha}_{\omega}(G) \ge \tfrac{1}{2}|V(G)|$, thereby establishing $^{\alpha}_{\omega}(\mathcal{K})=\tfrac{1}{2}$ and settling two BCZ conjectures. It also proves the same ratio for subcubic graphs, with a simple partition-based construction showing tightness. The methods hinge on decomposing $K_4$-minor-free graphs into leaf-blocks of series-parallel type and using contra-pairs to exclude counterexamples. The results deepen understanding of indeque structures in sparse graph classes and highlight connections to planarity and tree-width 2 graphs. Potential directions include extending bounds to broader classes with bounded maximum average degree and exploring computational complexity boundaries.

Abstract

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 $\frac 1 2$.

Figures (3)

• Figure 1: A series composition (Left) and a parallel composition (Right).
• Figure 2: All possible 0-parallel-pieces with $v$ (edges in dashed lines may or may not be present).
• Figure 3: A triangle-string (Left) and a kite (Right).

Theorems & Definitions (14)

• Definition 1.1
• Theorem 1.2: Biro, Collado and Zamora, BCZ
• Conjecture 1.3: Biro, Collado and Zamora, BCZ
• Conjecture 1.4: Biro, Collado and Zamora, BCZ
• Theorem 1.5
• Theorem 2.1: Duffin D
• Theorem 2.2
• Lemma 2.3: Eppstein, DE
• Corollary 2.4
• Lemma 2.5