The Tessellation Cover Number of Good Tessellable Graphs
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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 t-tessellability problem aims to decide whether a graph G has T(G)≤ t and is NP-complete for t≥ 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 good tessellable recognition (gtr) problem aims to decide whether G is a good tessellable graph. We show that gtr is 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.
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