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22
Asteroidal TripleFree Graphs
, 1997
"... . An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triples. The motivation for this investigation was provided, in ..."
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Cited by 55 (10 self)
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. An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triples. The motivation for this investigation was provided, in part, by the fact that the asteroidal triplefree graphs provide a common generalization of interval, permutation, trapezoid, and cocomparability graphs. The main contribution of this work is to investigate and reveal fundamental structural properties of ATfree graphs. Specifically, we show that every connected ATfree graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. We then provide characterizations of ATfree graphs in terms of dominating pairs and minimal triangulations. Subsequently, we state and prove a decomposition theorem for ATfree graphs. An assortment of other properties of ATfree graphs is also p...
Fixed parameter algorithms for planar dominating set and related problems
, 2000
"... We present an algorithm that constructively produces a solution to the kdominating set problem for planar graphs in time O(c √ kn), where c = 36√34. To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O ( � γ(G)), and that such a tree decomposition ca ..."
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Cited by 35 (10 self)
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We present an algorithm that constructively produces a solution to the kdominating set problem for planar graphs in time O(c √ kn), where c = 36√34. To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O ( � γ(G)), and that such a tree decomposition can be found in O ( � γ(G)n) time. The same technique can be used to show that the kface cover problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved √ k in O(c1 n + n2) time, where c1 = 236√34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of kdominating set, e.g., kindependent dominating set and kweighted dominating set. Keywords. NPcomplete problems, fixed parameter tractability, planar graphs, planar dominating set, face cover, outerplanarity, treewidth.
Linear Time Algorithms for Dominating Pairs in Asteroidal Triplefree Graphs
 SIAM J. Comput
, 1997
"... An independent set of three of vertices is called an asteroidal triple if between each pair in the triple there exists a path that avoids the neighbourhood of the third. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triple. The motivation for this work is pro ..."
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Cited by 25 (7 self)
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An independent set of three of vertices is called an asteroidal triple if between each pair in the triple there exists a path that avoids the neighbourhood of the third. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triple. The motivation for this work is provided, in part, by the fact that ATfree graphs offer a common generalization of interval, permutation, trapezoid, and cocomparability graphs. Previously, the authors have given an existential proof of the fact that every connected ATfree graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. The main contribution of this paper is a constructive proof of the existence of dominating pairs in connected ATfree graphs. The resulting simple algorithm, based on the wellknown Lexicographic BreadthFirst Search, can be implemented to run in time linear in the size of the input, whereas the best algorithm previousl...
Minimal triangulations of graphs: A survey
 Discrete Mathematics
"... Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was ..."
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Cited by 25 (3 self)
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Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was first studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we know that minimal triangulations are closely related to minimal separators of the input graph. Since the first papers presenting minimal triangulation algorithms appeared in 1976, several characterizations of minimal triangulations have been proved, and a variety of algorithms exist for computing minimal triangulations of both general and restricted graph classes. This survey presents and ties together these results in a unified modern notation, keeping an emphasis on the algorithms. 1 Introduction and
A Linear Time Algorithm to Compute a Dominating Path in an ATfree Graph
 Inform. Process. Lett
, 1998
"... An independent set fx; y; zg is called an asteroidal triple if between any pair in the triple there exists a path that avoids the neighborhood of the third. A graph is referred to as ATfree if it does not contain an asteroidal triple. We present a simple lineartime algorithm to compute a domina ..."
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Cited by 13 (3 self)
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An independent set fx; y; zg is called an asteroidal triple if between any pair in the triple there exists a path that avoids the neighborhood of the third. A graph is referred to as ATfree if it does not contain an asteroidal triple. We present a simple lineartime algorithm to compute a dominating path in a connected ATfree graph. Keywords. asteroidal triplefree graphs, domination, algorithms 1 Introduction A number of families of graphs including interval graphs [10], permutation graphs [6], trapezoid graphs [3, 5], and cocomparability graphs [8] feature a type of linear ordering of their vertex sets. It is precisely this linear ordering that is exploited in a search for efficient algorithms on these classes of graphs [2, 5, 7, 8, 9, 11, 12]. As it turns out, the classes mentioned above are all subfamilies of a class of graphs called the asteroidal triplefree graphs (ATfree graphs, for short). An independent triple fx; y; zg is called an asteroidal triple if between any p...
Domination and Total Domination on Asteroidal TripleFree Graphs
, 1996
"... We present the first polynomial time algorithms for solving the NPcomplete graph problems DOMINATING SET and TOTAL DOMINATING SET when restricted to asteroidal triplefree graphs. We also present algorithms to compute a minimum cardinality dominating set and a minimum cardinality total dominating s ..."
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Cited by 6 (2 self)
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We present the first polynomial time algorithms for solving the NPcomplete graph problems DOMINATING SET and TOTAL DOMINATING SET when restricted to asteroidal triplefree graphs. We also present algorithms to compute a minimum cardinality dominating set and a minimum cardinality total dominating set on a large superclass of the asteroidal triplefree graphs, called DDPgraphs. These algorithms can be implemented to run in time O(n 6 ) on asteroidaltriple free graphs and in time O(n 7 ) on DDPgraphs. 1 Introduction Asteroidal triplefree graphs (short ATfree graphs) form a large class of graphs containing interval, permutation, trapezoid and cocomparability graphs. Since 1989 ATfree graphs have been studied extensively by Corneil, Olariu and Stewart. They have published a collection of papers presenting many structural and algorithmic properties of ATfree graphs (see e.g. [1013]). By now the knowledge on the algorithmic complexity of NPcomplete graph problems when restri...
On the algorithmic complexity of twelve covering and independence parameters of graphs
 Discrete Applied Mathematics
, 1999
"... The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning ..."
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Cited by 4 (1 self)
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The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned twelve covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NPcompleteness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs.
Kayles and Nimbers
, 2000
"... Kayles is a combinatorial game on graphs. Two players select alternatingly a vertex from a given graph G  a chosen vertex may not be adjacent or equal to an already chosen vertex. The last player that can select a vertex wins the game. The problem to determine which player has a winning strategy is ..."
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Cited by 4 (1 self)
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Kayles is a combinatorial game on graphs. Two players select alternatingly a vertex from a given graph G  a chosen vertex may not be adjacent or equal to an already chosen vertex. The last player that can select a vertex wins the game. The problem to determine which player has a winning strategy is known to be PSPACEcomplete. Because of certain characteristics of the Kayles game, it can be analyzed with SpragueGrundy theory. In this way, we can show that the problem is polynomial time solvable for graphs with a bounded asteroidal number. It is shown that the problem can be solved in O(n^3) time on cocomparability graphs and circular arc graphs, and in O(n 1 1/ log 3) = O(n^1.631) time on cographs.
Computing a Dominating Pair in an Asteroidal Triplefree Graph in Linear Time
 in Algorithms and Data Structures WADS '95, Lecture
, 1998
"... An independent set of three of vertices is called an asteroidal triple if between each pair in the triple there exists a path that avoids the neighborhood of the third. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triple. The motivation for this work is prov ..."
Abstract

Cited by 3 (2 self)
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An independent set of three of vertices is called an asteroidal triple if between each pair in the triple there exists a path that avoids the neighborhood of the third. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triple. The motivation for this work is provided, in part, by the fact that ATfree graphs offer a common generalization of interval, permutation, trapezoid, and cocomparability graphs. Previously, the authors have given an existential proof of the fact that every connected ATfree graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. The main contribution of this paper is a constructive proof of the existence of dominating pairs in connected ATfree graphs. The resulting simple algorithm can be implemented to run in time linear in the size of the input, whereas the best algorithm previously known for this problem has complexity O(jV j 3 ) for input...