Recognizing Interval Digraphs and Interval Bigraphs in Polynomial Time

Abstract An interval bigraph is an undirected bipartite graph whose edge set is the intersection of the edge sets of an interval graph and the edge set of a complete bipartite graph on the same vertex set. A bipartite interval representation of an interval bigraph is given by a bipartitioned set of intervals for its vertices, such that vertices are adjacent if and only if the corresponding intervals intersect and belong to opposite sides of the bipartition. Interval digraphs are directed graphs defined by a closely related concept. Each vertex of an interval digraph is represented by two intervals on the real line, a source interval and a target interval. The directed arc ( u , v ) exists in the interval digraph if the source interval of u meets the target interval of v . We give a dynamic programming algorithm recognizing interval bigraphs (interval digraphs) in polynomial time. This algorithm recursively constructs a bipartite interval representation of a graph from bipartite interval representations of proper subgraphs. Moreover, we list some forbidden substructures of interval bigraphs.

[1]  Jitender S. Deogun,et al.  On Vertex Ranking for Permutations and Other Graphs , 1994, STACS.

[2]  Andreas Brandstiidt.  Classes of bipartite graphs related to chordal graphs , 1991 .

[3]  C. Lekkeikerker,et al.  Representation of a finite graph by a set of intervals on the real line , 1962 .

[4]  P. Duchet.  Classical Perfect Graphs: An introduction with emphasis on triangulated and interval graphs , 1984 .

[5]  Jeremy P. Spinrad,et al.  Bipartite permutation graphs , 1987, Discret. Appl. Math..

[6]  Mihalis Yannakakis,et al.  Node-Deletion Problems on Bipartite Graphs , 1981, SIAM J. Comput..

[7]  Edward Szpilrajn-Marczewski.  Sur deux propriétés des classes d'ensembles , 1945 .

[8]  Douglas B. West,et al.  Interval digraphs: An analogue of interval graphs , 1989, J. Graph Theory.

[9]  Christina Zamfirescu,et al.  Connection digraphs and second-order line digraphs , 1982, Discret. Math..

[10]  F. Gavril.  The intersection graphs of subtrees in tree are exactly the chordal graphs , 1974 .

[11]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .

[12]  Malay K. Sen,et al.  Indifference Digraphs: A Generalization of Indifference Graphs and Semiorders , 1994, SIAM J. Discret. Math..

[13]  F. Glover.  Maximum matching in a convex bipartite graph , 1967 .

[14]  George Steiner.  The recognition of indifference digraphs and generalized semiorders , 1996, J. Graph Theory.

[15]  Dieter Kratsch,et al.  Treewidth and Pathwidth of Permutation Graphs , 1995, SIAM J. Discret. Math..

[16]  Martin Charles Golumbic,et al.  Comparability graphs and a new matroid , 1977, J. Comb. Theory, Ser. B.

[17]  Ryan B. Hayward,et al.  Weakly triangulated graphs , 1985, J. Comb. Theory B.

[18]  Edward R. Scheinerman.  Characterizing intersection classes of graphs , 1985, Discret. Math..

[19]  Dieter Kratsch,et al.  Computing a Perfect Edge Without Vertex Elimination Ordering of a Chordal Bipartite Graph , 1995, Inf. Process. Lett..

[20]  Martin Charles Golumbic,et al.  Perfect Elimination and Chordal Bipartite Graphs , 1978, J. Graph Theory.

[21]  Martin Farber,et al.  Characterizations of strongly chordal graphs , 1983, Discret. Math..

[22]  G. Dirac.  On rigid circuit graphs , 1961 .