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- W3208343810 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 Tt(G) is the size of a minimum total tessellation cover of G. We present lower bounds Tt(G)≥ω(G) and Tt(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 Tt(G)=ω(G) or Tt(G)=s(G)+1. We study the complexity of the k-total tessellability problem, which aims to decide whether a given graph G has Tt(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." @default.
- W3208343810 created "2021-11-08" @default.
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- W3208343810 date "2022-12-01" @default.
- W3208343810 modified "2023-10-16" @default.
- W3208343810 title "Total tessellation cover: Bounds, hardness, and applications" @default.
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- W3208343810 doi "https://doi.org/10.1016/j.dam.2021.09.032" @default.
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