Article
On the PathWidth of Planar Graphs.
SIAM J. Discrete Math 01/2009; 23:13111316. DOI: 10.1137/060670146
Source: DBLP

Article: Pathwidth of outerplanar graphs
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ABSTRACT: We are interested in the relation between the pathwidth of a biconnected outerplanar graph and the pathwidth of its (geometric) dual. Bodlaender and Fomin, after having proved that the pathwidth of every biconnected outerplanar graph is always at most twice the pathwidth of its (geometric) dual plus two, conjectured that there exists a constant $c$ such that the pathwidth of every biconnected outerplanar graph is at most $c$ plus the pathwidth of its dual. They also conjectured that this was actually true with $c$ being $1$ for every biconnected planar graph. Fomin proved that the second conjecture is true for all planar triangulations, and made a stronger conjecture about the linear width of planar graphs. First, we construct for each p>=1 a biconnected outerplanar graph of pathwidth 2p+1 whose (geometric) dual has pathwidth p+1, thereby disproving all three conjectures. Then we prove, in an algorithmic way, that the pathwidth of every biconnected outerplanar graph is at most twice the pathwidth of its (geometric) dual minus 1. A tight interval for the studied relation is therefore obtained, and we show that all the gaps within the interval actually happen.Journal of Graph Theory 01/2007; · 0.63 Impact Factor 
Conference Paper: Nondeterministic Graph Searching: From Pathwidth to Treewidth.
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ABSTRACT: We introduce nondeterministic graph searching with a controlled amount of nondeterminism and show how this new tool can be used in algorithm design and combinatorial analysis applying to both pathwidth and treewidth. We prove equivalence between this gametheoretic approach and graph decompositions called q branched tree decompositions, which can be interpreted as a parameterized version of tree decompositions. Path decomposition and (standard) tree decomposition are two extreme cases of qbranched tree decompositions. The equivalence between nondeterministic graph searching and qbranched tree decomposition enables us to design an exact (exponential time) algorithm computing qbranched treewidth for all q≥0, which is thus valid for both treewidth and pathwidth. This algorithm performs as fast as the best known exact algorithm for pathwidth. Conversely, this equivalence also enables us to design a lower bound on the amount of nondeterminism required to search a graph with the minimum number of searchers.Mathematical Foundations of Computer Science 2005, 30th International Symposium, MFCS 2005, Gdansk, Poland, August 29  September 2, 2005, Proceedings; 01/2005  [Show abstract] [Hide abstract]
ABSTRACT: A graph parameter is selfdual in some class of graphs embeddable in some surface if its value does not change in the dual graph more than a constant factor. Selfduality has been examined for several widthparameters, such as branchwidth, pathwidth, and treewidth. In this paper, we give a direct proof of the selfduality of branchwidth in graphs embedded in some surface. In this direction, we prove that bw(G ) 6 bw(G) + 2g 4 for any graph G embedded in a surface of Euler genus g.Discrete Applied Mathematics. 01/2011; 159:21842186.
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