好可镶嵌图的镶嵌覆盖数
The Tessellation Cover Number of Good Tessellable Graphs
摘要：
一个图的镶嵌是一个划分的顶点到顶点不相交的团。图的镶嵌覆盖是覆盖其所有边的镶嵌的集合，并且镶嵌覆盖数，由$T(G)$表示，是最小镶嵌覆盖的大小。\textsc{$t$-tessellability}问题的目的是判定图$G$是否具有$T(G)\leq t$，并且对于$t\geq 3$是$\mathcal{NP}$ -完全的.由于$G$的最大诱导星的边数是$T(G)$的一个下界，我们定义好的可镶嵌图为~ $G$使得$T(G)=is(G)$ . \textsc{good tessellable recognition (gtr)}问题的目的是决定$G$是否是一个好的可镶嵌图。我们证明了\textsc{gtr}是$\mathcal{NP}$ -完全的，不仅当$T(G)$是已知的或$is(G)$是固定的，而且当$T(G)$和$is(G)$之间的差距是大的。作为一个副产品，我们得到的图形类，服从相应的计算复杂性行为。
英文摘要：
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 \textsc{$t$-tessellability} problem aims 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 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 \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, 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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