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A Width Parameter Useful for Chordal and Co-comparability Graphs

  • Dong Yeap Kang
  • O-joung Kwon
  • Torstein J. F. Strømme
  • Jan Arne Telle
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10167)

Abstract

In 2013 Belmonte and Vatshelle used mim-width, a graph parameter bounded on interval graphs and permutation graphs that strictly generalizes clique-width, to explain existing algorithms for many domination-type problems, known as LC-VSVP problems. We focus on chordal graphs and co-comparability graphs, that strictly contain interval graphs and permutation graphs respectively. First, we show that mim-width is unbounded on these classes, thereby settling an open problem from 2012. Then, we introduce two graphs \(K_t \boxminus K_t\) and \(K_t \boxminus S_t\) to restrict these graph classes, obtained from the disjoint union of two cliques of size t, and one clique of size t and one independent set of size t respectively, by adding a perfect matching. We prove that \((K_t \boxminus S_t)\)-free chordal graphs have mim-width at most \(t-1\), and \((K_t \boxminus K_t)\)-free co-comparability graphs have mim-width at most \(t-1\). From this, we obtain several algorithmic consequences, for instance, while Dominating Set is NP-complete on chordal graphs, it can be solved in time \(\mathcal {O}(n^{t})\) on chordal graphs where t is the maximum among induced subgraphs \(K_t \boxminus S_t\) in the given graph. We also show that classes restricted in this way have unbounded rank-width which validates our approach.

In the second part, we generalize these results to bigger classes. We introduce a new width parameter sim-width, special induced matching-width, by making only a small change in the definition of mim-width. We prove that chordal and co-comparability graphs have sim-width at most 1. Since Dominating Set is NP-complete on chordal graphs, an XP algorithm parameterized only by sim-width would imply P = NP. Therefore, to apply the algorithms for domination-type problems mentioned above, we parameterize by both sim-width w and a further parameter t, which is the smallest value such that the input has no induced minor isomorphic to \(K_t \boxminus S_t\) or \(K_t \boxminus S_t\). We show that such graphs have mim-width at most \(8(w+1)t^3\) and that the resulting algorithms for domination-type problems have runtime \(n^{\mathcal {O}(wt^3)}\), when the decomposition tree is given.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Dong Yeap Kang
    • 1
  • O-joung Kwon
    • 2
  • Torstein J. F. Strømme
    • 3
  • Jan Arne Telle
    • 3
  1. 1.Department of Mathematical SciencesKAISTDaejeonSouth Korea
  2. 2.Logic and SemanticsTU BerlinBerlinGermany
  3. 3.Department of InformaticsUniversity of BergenBergenNorway

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