The Tessellation Cover Number of Good Tessellable Graphs

TL;DR

This paper investigates the computational complexity of recognizing good tessellable graphs, where the tessellation cover number equals the maximum induced star size, proving NP-completeness in various scenarios.

Contribution

It introduces the concept of good tessellable graphs and proves that recognizing them is NP-complete, even with large gaps between key parameters.

Findings

NP-completeness of the recognition problem for good tessellable graphs

Complexity persists when the tessellation cover number is known or the maximum induced star size is fixed

Graph classes with specific complexity behaviors are identified

Abstract

A tessellation of a graph is a partition of its vertices into vertex disjoint cliques. A tessellation cover of a graph is a set of tessellations that covers all of its edges, and the tessellation cover number, denoted by $T(G)$, is the size of a smallest tessellation cover. The \textsc{$t$-tessellability} problem aims to decide whether a graph $G$ has $T(G)\leq t$ and is $\mathcal{NP}$-complete for $t\geq 3$. Since the number of edges of a maximum induced star of $G$, denoted by $is(G)$, is a lower bound on $T(G)$, we define good tessellable graphs as the graphs~$G$ such that $T(G)=is(G)$. The \textsc{good tessellable recognition (gtr)} problem aims to decide whether $G$ is a good tessellable graph. We show that \textsc{gtr} is $\mathcal{NP}$-complete not only if $T(G)$ is known or $is(G)$ is fixed, but also when the gap between $T(G)$ and $is(G)$ is large. As a byproduct, we obtain graph classes that obey the corresponding computational complexity behaviors.