On the Path-Width of Planar Graphs
SIAM Journal on Discrete Mathematics (Impact Factor: 0.65). 01/2009; 23(3):1311-1316. DOI: 10.1137/060670146
In this paper, we present a result concerning the relation between the path-with of a planar graph and the path-width of its dual. More precisely, we prove that for a 3-connected planar graph G, pw(G) 3pw(G ) + 2. For 4-connected planar graphs, and more generally for Hamiltonian planar graphs, we prove a stronger bound pw(G ) 2 pw(G) + c. The best previously known bound was obtained by Fomin and Thilikos who proved that pw(G ) 6 pw(G) + cte. The proof is based on an algorithm which, given a xed spanning tree of G, transforms any given decomposition of G into one of G. The ratio of the corresponding parameters is bounded by the maximum degree of the spanning tree.
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Conference Paper: Nondeterministic Graph Searching: From Pathwidth to Treewidth[Show abstract] [Hide abstract]
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 game-theoretic 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 q-branched tree decompositions. The equivalence between nondeterministic graph searching and q-branched tree decomposition enables us to design an exact (exponential time) algorithm computing q-branched 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.
Article: Path-width of Outerplanar Graphs[Show abstract] [Hide abstract]
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.