## Pathwidth, Bandwidth and Completion Problems to Proper Interval Graphs with Small Cliques (1996)

Venue: | SIAM Journal on Computing |

Citations: | 29 - 6 self |

### BibTeX

@ARTICLE{Kaplan96pathwidth,bandwidth,

author = {Haim Kaplan and Ron Shamir},

title = {Pathwidth, Bandwidth and Completion Problems to Proper Interval Graphs with Small Cliques},

journal = {SIAM Journal on Computing},

year = {1996},

volume = {25},

pages = {540--561}

}

### OpenURL

### Abstract

We study two related problems motivated by molecular biology: ffl Given a graph G and a constant k, does there exist a supergraph G of G which is a unit interval graph and has clique size at most k? ffl Given a graph G and a proper k-coloring c of G, does there exist a supergraph We show that those problems are polynomial for fixed k. On the other hand we prove that the first problem is equivalent to deciding if the bandwidth of G is at most k \Gamma 1. Hence, it is NP-hard, and W [t]-hard for all t. We also show that the second problem is W [1]-hard. This implies that for fixed k, both of the problems are unlikely to have an O(n ) algorithm, where ff is a constant independent of k.