Boolean-width of graphs

We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods–Boolean sums of neighborhoods–across a cut of a graph. For many graph problems, this number is the runtime bottleneck when using a divide-and-conquer approach. For an n -vertex graph given with...

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Vydáno v:Theoretical computer science Ročník 412; číslo 39; s. 5187 - 5204
Hlavní autoři: Bui-Xuan, Binh-Minh, Telle, Jan Arne, Vatshelle, Martin
Médium: Journal Article
Jazyk:angličtina
Vydáno: Oxford Elsevier B.V 09.09.2011
Elsevier
Témata:
Algorithmics. Computability. Computer arithmetics
Applied sciences
Boolean algebra
Combinatorics. Ordered structures
Computer Science
Computer science; control theory; systems
Counting
Data Structures and Algorithms
Decomposition
Exact sciences and technology
FPT algorithm
Graph decomposition
Graphs
Information retrieval. Graph
Mathematics
Minimum weight
Miscellaneous
Order, lattices, ordered algebraic structures
Sciences and techniques of general use
Sums
Theoretical computing
Trees
Unions
Width parameter
Graph decomposition
FPT algorithm
Boolean algebra
Width parameter
Bottleneck
Clique width
Grid pattern
Dominating set
Computer theory
Graph clique
Vertex(graph)
Rank
Algorithm
Minimum time
Tree width
Graph cut
Independent set
ISSN:0304-3975, 1879-2294
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Popis
Shrnutí:We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods–Boolean sums of neighborhoods–across a cut of a graph. For many graph problems, this number is the runtime bottleneck when using a divide-and-conquer approach. For an n -vertex graph given with a decomposition tree of boolean-width k , we solve Maximum Weight Independent Set in time O ( n 2 k 2 2 k ) and Minimum Weight Dominating Set in time O ( n 2 + n k 2 3 k ) . With an additional n 2 factor in the runtime, we can also count all independent sets and dominating sets of each cardinality. Boolean-width is bounded on the same classes of graphs as clique-width. boolean-width is similar to rank-width, which is related to the number of G F ( 2 ) -sums of neighborhoods instead of the Boolean sums used for boolean-width. We show for any graph that its boolean-width is at most its clique-width and at most quadratic in its rank-width. We exhibit a class of graphs, the Hsu-grids, having the property that a Hsu-grid on Θ ( n 2 ) vertices has boolean-width Θ ( log n ) and rank-width, clique-width, tree-width, and branch-width Θ ( n ) .
Bibliografie:ObjectType-Article-2
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ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2011.05.022