On the Computational Complexity of the Forcing Chromatic Number

Let χ(G) denote the chromatic number of a graph G. A colored set of vertices of G is called forcing if its coloring is extendable to a proper χ(G)-coloring of the whole graph in a unique way. The forcing chromatic number Fχ(G) is the smallest cardinality of a forcing set of G. We estimate the computational complexity of Fχ(G) relating it to the complexity class US introduced by Blass and Gurevich. We prove that recognizing if Fχ(G) ≤ 2 is US-hard with respect to polynomial-time many-one reductions. Furthermore, this problem is coNP-hard even under the promises that Fχ(G) ≤ 3 and G is 3-chromatic. On the other hand, recognizing if Fχ(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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