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 χ (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 F χ (G) relating it to the complexity class US introduced by Blass and Gurevich. We prove that recognizing if F χ (G) ≤ 2 is US-hard with respect to polynomial-time many-one reductions. Moreover, this problem is coNP-hard even under the promises that F χ (G) ≤ 3 and G is 3-chromatic. On the other hand, recognizing if F χ (G) ≤ 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.

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