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A tessellation of a graph is a partition of its vertices into vertex disjointcliques. A tessellation cover of a graph is a set of tessellations that coversall of its edges, and the tessellation cover number, denoted by $T(G)$, is thesize of a smallest tessellation cover. The \textsc{$t$-tessellability} problemaims to decide whether a graph $G$ has $T(G)\leq t$ and is$\mathcal{NP}$-complete for $t\geq 3$. Since the number of edges of a maximuminduced star of $G$, denoted by $is(G)$, is a lower bound on $T(G)$, we definegood tessellable graphs as the graphs~$G$ such that $T(G)=is(G)$. The\textsc{good tessellable recognition (gtr)} problem aims to decide whether $G$is a good tessellable graph. We show that \textsc{gtr} is$\mathcal{NP}$-complete not only if $T(G)$ is known or $is(G)$ is fixed, butalso when the gap between $T(G)$ and $is(G)$ is large. As a byproduct, weobtain graph classes that obey the corresponding computational complexitybehaviors.

The Tessellation Cover Number of Good Tessellable Graphs