Permutation bigraphs and interval containments

Saha, P. K.; Basu, Asim; Sen, M. K.; West, Douglas B.

doi:10.1016/j.dam.2014.05.020

Cited by 7 publications

(3 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will use this equivalence in Section 3. The other equivalent classes can be found in [16,22]. Among those, we choose the model of interval containment bigraphs because of the simplicity of the construction of 12-representants.…”

Section: Necessary Conditionmentioning

confidence: 99%

Graph classes equivalent to 12-representable graphs

Takaoka¹

2022

Preprint

View full text Add to dashboard Cite

Jones et al. (2015) introduced the notion of u-representable graphs, where u is a word over {1, 2} different from 22 • • • 2, as a generalization of word-representable graphs. Kitaev (2016) showed that if u is of length at least 3, then every graph is u-representable. This indicates that there are only two nontrivial classes in the theory of u-representable graphs: 11-representable graphs, which correspond to word-representable graphs, and 12-representable graphs. This study deals with 12-representable graphs.Jones et al. ( 2015) provided a characterization of 12-representable trees in terms of forbidden induced subgraphs. Chen and Kitaev (2022) presented a forbidden induced subgraph characterization of a subclass of 12-representable grid graphs.This paper shows that a bipartite graph is 12-representable if and only if it is an interval containment bigraph. The equivalence gives us a forbidden induced subgraph characterization of 12-representable bipartite graphs since the list of minimal forbidden induced subgraphs is known for interval containment bigraphs. We then have a forbidden induced subgraph characterization for grid graphs, which solves an open problem of Chen and Kitaev (2022). The study also shows that a graph is 12-representable if and only if it is the complement of a simple-triangle graph. This equivalence indicates that a necessary condition for 12-representability presented by Jones et al. (2015) is also sufficient. Finally, we show from these equivalences that 12-representability can be determined in O(n 2 ) time for bipartite graphs and in O(n( m + n)) time for arbitrary graphs, where n and m are the number of vertices and edges of the complement of the given graph.

show abstract

Section: Necessary Conditionmentioning

confidence: 99%

Graph classes equivalent to 12-representable graphs

Takaoka¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Interval containment bigraphs [14] can be considered a bipartite analogue of permutation graphs [15] because they are equivalent to containment graphs of intervals. The class of interval containment bigraphs is equivalent to other wellinvestigated classes, e.g., bipartite graphs whose complements are circular-arc graphs [4], two-directional orthogonal ray graphs [21], and permutation bigraphs [20], see also [23]. Note that the permutation bigraphs [20] differ from bipartite permutation graphs studied in [22].…”

Section: Introductionmentioning

confidence: 99%

“…The class of interval containment bigraphs is equivalent to other wellinvestigated classes, e.g., bipartite graphs whose complements are circular-arc graphs [4], two-directional orthogonal ray graphs [21], and permutation bigraphs [20], see also [23]. Note that the permutation bigraphs [20] differ from bipartite permutation graphs studied in [22]. It has been pointed out recently that interval containment bigraphs can also be viewed as a bipartite analogue of interval graphs since they share many nice properties [11], [12].…”

Section: Introductionmentioning

confidence: 99%

Decomposition of <i>P</i><sub>6</sub>-Free Chordal Bipartite Graphs

Takaoka

2023

IEICE Trans. Fundamentals

View full text Add to dashboard Cite

Canonical decomposition for bipartite graphs, which was introduced by Fouquet, Giakoumakis, and Vanherpe (1999), is a decomposition scheme for bipartite graphs associated with modular decomposition. Weak-bisplit graphs are bipartite graphs totally decomposable (i.e., reducible to single vertices) by canonical decomposition. Canonical decomposition comprises series, parallel, and K + S decomposition. This paper studies a decomposition scheme comprising only parallel and K + S decomposition. We show that bipartite graphs totally decomposable by this decomposition are precisely P 6 -free chordal bipartite graphs. This characterization indicates that P 6 -free chordal bipartite graphs can be recognized in linear time using the recognition algorithm for weak-bisplit graphs presented by Giakoumakis and Vanherpe (2003).

show abstract