Partitioning a graph into $Δ$-convex sets of graphs and graph products

TL;DR

This work studies partitioning graphs into $\Delta$-convex sets, where a set $S$ is $\Delta$-convex if the Δ-interval $[S]$ equals $S$. It establishes NP-completeness for Convex $p$-cover and Convex $p$-partition for every fixed $p \ge 4$ in $\Delta$-convexity, and provides a polynomial-time $O(n^3)$ algorithm for Convex $2$-Partition, using a reduction from $k$-Colorability. The paper then analyzes three standard graph products—Cartesian, strong, and lexicographic—deriving exact or bounded values for the convex cover/partition numbers in various cases (notably when $\Delta$-extreme vertices or cut vertices exist) and showing reductions to the chromatic number in certain products. Overall, it maps the complexity landscape of $\Delta$-convex decompositions and product behavior, offering precise product-specific results and useful bounds that connect to hull numbers and chromatic properties.

Abstract

Given a graph $G$ and a set $S \subseteq V(G)$, we say that $S$ is $Δ$-convex if the neighborhood of every vertex not in $S$ is an independent set. A collection ${\cal V} = (V_1, V_2, \ldots , V_p)$ of convex sets of $G$ is a convex $p$-cover if $V(G) = \underset{1 \leq i \leq p}{\bigcup} V_i$ and $V_i \nsubseteq {\underset{1 \leq j \leq p, j\ne i}{\bigcup}} V_j$ for $i \in \{1, \ldots, p\}$. If the convex sets of ${\cal V}$ are pairwise disjoint, ${\cal V}$ is a convex $p$-partition of $V(G)$. The convex cover number $φ_c(G)$ (the convex partition number $Θ_c(G)$) of a graph $G$ is the least integer $p \geq 2$ for which $G$ has a convex $p$-cover (convex $p$-partition). In this work, we prove that the {\sc Convex p-cover} and {\sc Convex p-Partition} problems are \NP-complete for any fixed $p \ge 4$ in $Δ$-convexity. Furthermore, for the three standard graph products, namely, the Cartesian, strong and lexicographic products, we determine these parameters for some cases and present bounds for others.

Figures (1)

• Figure 1: Graph $G$ such that $h(G) \ge 3$ and $\phi_c(G) \ge 3$. In order to make easier to understand, 10 edges are not drawn, namely, each of the 5 rectangles formed by dashed lines and 4 vertices is a $K_4$.

Theorems & Definitions (22)

• Theorem 1
• proof
• Theorem 2
• Corollary 3
• Lemma 4
• proof
• Theorem 5
• proof
• Theorem 6
• proof