Total tessellation cover: Bounds, hardness, and applications

Abstract

The concept of graph tessellation cover was defined in the context of quantum walk models, and is a current research area in graph theory. In this work, we propose a generalization called total tessellation cover. A tessellation of a graph G is a partition of its vertex set V ( G ) into vertex disjoint cliques. A tessellation cover of G is a set of tessellations that covers its edge set E ( G ) . A total tessellation cover of G consists of a tessellation cover together with a compatible vertex coloring, such that the color of each vertex is different from the tessellation labels of the edges incident to the vertex. The total tessellation cover number T t ( G ) is the size of a minimum total tessellation cover of G . We present lower bounds T t ( G ) ≥ ω ( G ) and T t ( G ) ≥ s ( G ) + 1 , where ω ( G ) is the size of a maximum clique, and s ( G ) is the number of edges of a maximum induced star subgraph. A graph G is called a good total tessellable if T t ( G ) = ω ( G ) or T t ( G ) = s ( G ) + 1 . We study the complexity of the k - total tessellability problem, which aims to decide whether a given graph G has T t ( G ) ≤ k . We prove that k - total tessellability is in P for good total tessellable graphs. We establish the NP -completeness of the k - total tessellability when restricted to the following graph classes: bipartite graphs, line graphs of triangle-free graphs, universal graphs, ( 2 , 1 ) -chordal graphs and planar graphs.