The <i>k</i>-Domination and <i>k</i>-Stability Problems on Sun-Free Chordal Graphs

Chang, Gerard J.; Nemhauser, George L.

doi:10.1137/0605034

Cited by 139 publications

(63 citation statements)

References 18 publications

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“…Lemma 1 [1] For any two vertices z and y of a chordal graph G and integer Such a vertex z* we will call a gate of a vertex z in C.…”

Section: Metric Properties Of Chordal Graphsmentioning

confidence: 99%

“…They were first introduced by Hajnal & Suranyi [10] and then studied extensively by many people, see [6,9] for general results. Recently, the metric properties of chordal graphs were also investigated, see [1][2][3][4][5][6][7]13,15]. In particular, the inequality 2r(G) _> d(G) >_ 2r(G) -2 was proven [1,3,4] and the: centers of chordal graphs were characterized [4].…”

Section: Introductionmentioning

confidence: 99%

“…Recently, the metric properties of chordal graphs were also investigated, see [1][2][3][4][5][6][7]13,15]. In particular, the inequality 2r(G) _> d(G) >_ 2r(G) -2 was proven [1,3,4] and the: centers of chordal graphs were characterized [4]. The class of chordal graphs contains trees, block graphs, maximal outerplanar graphs, k-trees, interval graphs and strongly chordal graphs.…”

Section: Introductionmentioning

confidence: 99%

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A linear-time algorithm for finding a central vertex of a chordal graph

Chepoi

Dragan

1994

Algorithms — ESA '94

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Abstract. In a graph G = (V, E), the eccentricity e(v) of a vertex v is max{d (v,u) : u E V}. The center of a graph is the set of vertices with minimum eccentricity. A graph G is chordal if every cycle of length at least four has a chord. We present an algorithm which computes in linear time a central vertex of a chordal graph. The algorithm uses the metric properties of chordal graphs and Tarjan and Yannakalds linear-time test for graph chordality.

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“…Lemma 1 [1] For any two vertices z and y of a chordal graph G and integer Such a vertex z* we will call a gate of a vertex z in C.…”

Section: Metric Properties Of Chordal Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A linear-time algorithm for finding a central vertex of a chordal graph

Chepoi

Dragan

1994

Algorithms — ESA '94

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“…The domination number of G, denoted by y(G), is the size of a smallest dominating set of G. We show that RRAF is .K.9-complete, by showing that this· problem is equivalent to the problem of finding minimum dominating sets in bipartite graphs, which is .K.9-complete (Rertossi [2], Chang and Nemhauser [3]). …”

Section: Rrafmentioning

confidence: 99%

Right angle free subsets in the plane

Gamble

Pulleyblank

Reed

et al. 1995

Graphs and Combinatorics

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Abstract. In this paper we investigate the complexity of finding maximum right angle free subsets of a given set of poi1.1-ts in the plane. For a set of rational points P in the plane. the right angle number p(P) (respectively rectilmear right angle number P.R(P)) of P is the cardinality of a maximum subset of P, no three members of which form a right angle triangle (respectively a right angle triangle with its side or base parallel to the x-axis~ It is shown that both parameters are .¥9'-hard to compute. The latter problem is also shown to be equivalent to finding a minimum dominating set in a bipartite graph. This is used to show that there is a polynomial algorithm for computing PB.(P) when P is a horizontally-convex subset of the lattice Z x Z (P is horizontally-convex if for any pair of points in P which lie on a horizontal line, every lattice point between them is also in P). We then show that this algorithm yields a !-approximate algorithm for the right angle number of a convex subregion of the lattice. t. IntroductionLet P be a set of points in the plane. A subset S of P is right angle free (nlf) if no three points of S form the vertices of a right angle triangle. The right angle number of P, denoted by p(P), is the cardinality of a maximum raf subset of P.

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“…Note that the decision problem DOMINATION remains NPcomplete if the instance is restricted to bipartite graphs (e.g., see [7]). …”

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confidence: 99%