Transactions on Combinatorics (Dec 2019)
Coloring problem of signed interval graphs
Abstract
A signed graph $(G,\sigma)$ is a graph together with an assignment of signs $\{+,-\}$ to its edges where $\sigma$ is the subset of its negative edges. There are a few variants of coloring and clique problems of signed graphs, which have been studied. An initial version known as vertex coloring of signed graphs is defined by Zaslavsky in $1982$. Recently Naserasr et. al., in [R. Naserasr, E. Rollova and E. Sopena, Homomorphisms of signed graphs, J. Graph Theory, 79 (2015) 178--212, have defined signed chromatic and signed clique numbers of signed graphs. In this paper we consider the latter mentioned problems for signed interval graphs. We prove that the coloring problem of signed interval graphs is NP-complete whereas their ordinary coloring problem (the coloring problem of interval graphs) is in P. Moreover we prove that the signed clique problem of a signed interval graph can be solved in polynomial time. We also consider the complexity of further related problems.
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