The Tessellation Cover Number of Good Tessellable Graphs
Alexandre Abreu, Lu\'is Cunha, Celina de Figueiredo, Luis Kowada,, Franklin Marquezino, Renato Portugal, Daniel Posner

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 , is the size of a smallest tessellation cover. The \textsc{-tessellability} problem aims to decide whether a graph has and is -complete for . Since the number of edges of a maximum induced star of , denoted by , is a lower bound on , we define good tessellable graphs as the graphs~ such that . The \textsc{good tessellable recognition (gtr)} problem aims to decide whether is a good tessellable graph. We show that \textsc{gtr} is -complete not only if is known or is fixed, but also when the gap between and is large. As a byproduct, we obtain…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph theory and applications · Graph Labeling and Dimension Problems
