Digraphs of bounded elimination width ✩

Hunter and Kreutzer recently introduced the digraph width measure Kelly-width. The measure is based on a vertex elimination process that aims at bounding the out-degree of a vertex during the elimination. We refine the Kelly-width measure by bounding both the out-degree and the in-degree of a vertex. We show that the elimination process and a subgraph characterisation of Kelly-width naturally generalise to the refined notion. The main result of the paper is a game characterisation of the refined measure, which is a surprisingly non-trivial generalisation of the game for Kelly-width.

[1]  Petr Hliněný,et al.  Are There Any Good Digraph Width Measures? , 2010, IPEC.

[2]  Dimitrios M. Thilikos,et al.  Fugitive-Search Games on Graphs and Related Parameters , 1997, Theor. Comput. Sci..

[3]  D. Rose,et al.  Algorithmic aspects of vertex elimination on directed graphs. , 1975 .

[4]  M. Yannakakis.  Computing the Minimum Fill-in is NP^Complete , 1981 .

[5]  Hans L. Bodlaender,et al.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC.

[6]  Stephan Kreutzer,et al.  DAG-Width and Parity Games , 2006, STACS.

[7]  Stephan Kreutzer,et al.  The dag-width of directed graphs , 2012, J. Comb. Theory, Ser. B.

[8]  Jan Arne Telle,et al.  Recognizing digraphs of Kelly-width 2 , 2010, Discret. Appl. Math..

[9]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs IV: Definability Properties of Equational Graphs , 1990, Ann. Pure Appl. Log..

[10]  Jan Obdrzálek,et al.  DAG-width: connectivity measure for directed graphs , 2006, SODA '06.

[11]  Jan Arne Telle,et al.  Chordal digraphs , 2012, Theor. Comput. Sci..

[12]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

[13]  Paul D. Seymour,et al.  Approximating clique-width and branch-width , 2006, J. Comb. Theory, Ser. B.

[14]  Peter L. Hammer,et al.  Discrete Applied Mathematics , 1993 .

[15]  Robert Ganian,et al.  Are there any good digraph width measures? , 2016, J. Comb. Theory, Ser. B.

[16]  Robin Thomas,et al.  Directed Tree-Width , 2001, J. Comb. Theory, Ser. B.

[17]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[18]  Derek G. Corneil,et al.  Complexity of finding embeddings in a k -tree , 1987 .

[19]  Stephan Kreutzer,et al.  Digraph decompositions and monotonicity in digraph searching , 2008, Theor. Comput. Sci..

[20]  Stephan Kreutzer,et al.  Digraph measures: Kelly decompositions, games, and orderings , 2007, SODA '07.

[21]  Bruno Courcelle,et al.  Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width , 1998, WG.

[22]  Stephan Kreutzer,et al.  Digraph Decompositions and Monotonicity in Digraph Searching , 2008, WG.

[23]  Donald J. Rose,et al.  Toward Characterization of Perfect Elimination Digraphs , 1973, SIAM J. Comput..