The graph tessellation cover number: extremal bounds, efficient algorithms and hardness

TL;DR

This paper investigates the tessellation cover number of graphs, providing bounds, identifying classes where these bounds are tight, and establishing complexity results including NP-completeness and a linear-time algorithm for specific cases.

Contribution

It introduces new bounds on the tessellation cover number, proves NP-completeness for various graph classes, and presents a linear-time algorithm for 2-tessellability.

Findings

Upper bounds based on chromatic index and clique graph chromatic number

NP-completeness for t-tessellability in several graph classes

Linear-time algorithm for 2-tessellability

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. The $t$-tessellability problem aims to decide whether there is a tessellation cover of the graph with $t$ tessellations. This problem is motivated by its applications to quantum walk models, in especial, the evolution operator of the staggered model is obtained from a graph tessellation cover. We establish upper bounds on the tessellation cover number given by the minimum between the chromatic index of the graph and the chromatic number of its clique graph and we show graph classes for which these bounds are tight. We prove $\mathcal{NP}$-completeness for $t$-tessellability if the instance is restricted to planar graphs, chordal (2,1)-graphs, (1,2)-graphs, diamond-free graphs with diameter five, or for any fixed $t$ at least 3. On the other hand, we improve the complexity for 2-tessellability to a linear-time algorithm.