A linear algorithm for the domination number of a series-parallel graph

Kikuno, Tohru; Yoshida, Noriyoshi; Kakuda, Yoshiaki

doi:10.1016/0166-218x(83)90003-3

Cited by 69 publications

(27 citation statements)

References 5 publications

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“…For (3), the class of outerplanar graphs is properly 3-colorable in linear time and closed under taking minors. Kikuno et al [22] present a linear time algorithm for finding a minimum dominating set in a series-parallel graph, which includes outerplanar graphs. The result follows by combining this linear time algorithm with the coloring algorithm mentioned above, but using just three colors instead of four.…”

Section: Approximability For Three or More Colorsmentioning

confidence: 99%

Three Colors Suffice: Conflict-Free Coloring of Planar Graphs

Abel¹,

Álvarez²,

Gour³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number χ CF (G) (the smallest k for which conflict-free k-colorings exist), with a focus on planar graphs.For general graphs, we prove the conflict-free variant of the famous Hadwiger Conjecture: If G does not contain K k+1 as a minor, then χ CF (G) ≤ k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. In addition, we give a complete characterization of the algorithmic/computational complexity of conflict-free coloring. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color, while for outerplanar graphs, this can be decided in polynomial time. Furthermore, it is NP-complete to decide whether a planar graph has a conflict-free coloring with two colors, while for outerplanar graphs, two colors always suffice. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs.

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Section: Approximability For Three or More Colorsmentioning

confidence: 99%

Three Colors Suffice: Conflict-Free Coloring of Planar Graphs

Abel¹,

Álvarez²,

Gour³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In this section, we propose a pseudo‐polynomial dynamic program (DP) that solves the BMCF( K ) problem on series‐parallel digraphs. We start with a formal definition of directed series‐parallel (SP) digraphs (similar to ) and then describe the DP. Definition Series‐parallel digraph An arc

a = (s, t)

is a SP‐digraph with a source s and a target t . Any digraph that is obtained by either a series operation or a parallel operation of two SP‐digraphs G 1 and G 2 is itself an SP‐digraph.…”

Section: Pseudo‐polynomially Solvable Casesmentioning

confidence: 99%

The budgeted minimum cost flow problem with unit upgrading cost

et al. 2016

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The budgeted minimum cost flow problem (BMCF(K)) with unit upgrading costs extends the classical minimum cost flow problem by allowing one to reduce the cost of at most K arcs. In this article, we consider complexity and algorithms for the special case of an uncapacitated network with just one source. By a reduction from 3‐SAT we prove strong N P ‐completeness and inapproximability, even on directed acyclic graphs. On the positive side, we identify three polynomially solvable cases: on arborescences, on so‐called tree‐like graphs, and on instances with a constant number of sinks. Furthermore, we develop dynamic programs with pseudo‐polynomial running time for the BMCF(K) problem on (directed) series‐parallel graphs and (directed) graphs of bounded treewidth. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 69(1), 67–82 2017

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“…(3) If (N(X) is of type S, then calculate a ( ( N ( X ) ) and r ( ( N ( X ) ) in a good order of parentheses that serially connected with its child series-parallel subnetworks.…”

Section: Inputmentioning

confidence: 99%

“…(2) If ( N ( X ) is of type P, then calculate a ( N ( X ) ) and r((N(X)) in an arbitrary order of parentheses that parallelly connected with its child seriesparallel subnetworks. (3) If (N(X) is of type S, then calculate a ( ( N ( X ) ) and r ( ( N ( X ) ) in a good order of parentheses that serially connected with its child series-parallel subnetworks.…”

Section: Inputmentioning

confidence: 99%

Maximum independent number for series‐parallel networks

Hsu

Wang

1991

Networks

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A series-parallel network is used as a model of electric circuits. Usually. we use a series-parallel graph to represent a series-parallel network. However, there are a lot of different graph representations for a single series-parallel network. For a given graph, a subset of V(G) is called an independent set if there are no edge of G joining two vertices in S. An independent set S of G is maximum if G has no independent set S' with IS'/ > IS[. The number of vertices in a maximum independent set of G is called the independent number of G and is denoted by a(G). The maximum independent number a ( N ) for a series-parallel network N is defined to be the maximum a(GIN]) among all possible graph representations C"J. In this paper, we are going to present a linear time algorithm to compute a ( N ) for any series-parallel network N.

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A linear algorithm for the domination number of a series-parallel graph

Cited by 69 publications

References 5 publications

Three Colors Suffice: Conflict-Free Coloring of Planar Graphs

Three Colors Suffice: Conflict-Free Coloring of Planar Graphs

The budgeted minimum cost flow problem with unit upgrading cost

Maximum independent number for series‐parallel networks

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