On the Computational Complexity of the Forcing Chromatic Number
<|reference_start|>On the Computational Complexity of the Forcing Chromatic Number: 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 $F(G)$ 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 $F(G)$ relating it to the complexity class US introduced by Blass and Gurevich. We prove that recognizing if $F(G)\le 2$ is US-hard with respect to polynomial-time many-one reductions. Moreover, this problem is coNP-hard even under the promises that $F(G)\le 3$ and $G$ is 3-chromatic. On the other hand, recognizing if $F(G)\le k$, for each constant $k$, is reducible to a problem in US via disjunctive truth-table reduction.   Similar results are obtained also for forcing variants of the clique and the domination numbers of a graph.<|reference_end|>
@article{harary2004on,
    title={On the Computational Complexity of the Forcing Chromatic Number},
    author={Frank Harary, Wolfgang Slany, Oleg Verbitsky},
    journal={arXiv preprint arXiv:cs/0406044},
    year={2004},
    archivePrefix={arXiv},
    eprint={cs/0406044},
    primaryClass={cs.CC}
}