Date: 03 Aug 2005

On the restriction of some NP-complete graph problems to permutation graphs

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Permutation graphs are known as a useful class of perfect graphs for which the NP-complete graph problems GRAPH k-COLORABILITY, PARTITION INTO CLIQUES, CLIQUE and INDEPENDENT SET (VERTEX COVER) (terminology from /8/) are solvable in polynomial time (/7/), in fact all four by the same algorithm (see /10/ for a presentation of these results).

In this paper we show that FEEDBACK VERTEX SET, MINIMUM NODE-DELETION BIPARTITE SUBGRAPH (and its generalization MINIMUM NODE-DELETION (k, l)-SUBGRAPH), and DOMINATING SET together with some variants (where the dominating set is also a clique or independent or connected or total or a path) are solvable in polynomial time when restricted to permutation graphs.

It is shown that each connected permutation graph has a dominating path i.e. a spanning tree which is a caterpillar with hair length 1. Furthermore we give a characterization of bipartite permutation graphs which as a byproduct also shows that such caterpillars are the only trees which are permutation graphs.

The characterization yields polynomial time solutions of HAMILTONIAN CIRCUIT and HAMILTONIAN PATH and also of a variant of CROSSING NUMBER on bipartite permutation graphs.