On the Computational Complexity of the Forcing Chromatic Number

Abstract

We consider vertex colorings of graphs in which adjacent vertices have distinct colors. A graph is $s$-chromatic if it is colorable in $s$ colors and any coloring of it uses at least $s$ colors. The forcing chromatic number $F_{\chi}(G)$ of an $s$-chromatic graph $G$ is the smallest number of vertices which must be colored so that, with the restriction that $s$ colors are used, every remaining vertex has its color determined uniquely. We estimate the computational complexity of $\force G$ relating it to the complexity class US introduced by Blass and Gurevich [Inform. Control, 55 (1982), pp. 80-88]. We prove that recognizing whether $F_{\chi}(G)\le2$ is US-hard with respect to polynomial-time many-one reductions. Moreover, this problem is coNP-hard even under the promises that $F_{\chi}(G)\le3$ and $G$ is 3-chromatic. On the other hand, recognizing whether $F_{\chi}(G)\le k$, for each constant $k$, is reducible to a problem in US via a disjunctive truth-table reduction. Similar results are obtained also for forcing variants of the clique and the domination numbers of a graph.