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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 ..."
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Cited by 12 (3 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(n⁶) on asteroidaltriple free graphs and in time O(n⁷) on DDPgraphs.
Convexity and HHDFree Graphs
, 1999
"... It is well known that chordal graphs can be characterized via<F3.302e+05><F3.821e+05> mconvexity. In this paper we introduce the notion of<F3.302e+05> m<F2.785e+05> 3<F3.821e+05> convexity (a relaxation of<F3.302e+05><F3.821e+05> mconvexity) which is cl ..."
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Cited by 12 (3 self)
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It is well known that chordal graphs can be characterized via<F3.302e+05><F3.821e+05> mconvexity. In this paper we introduce the notion of<F3.302e+05> m<F2.785e+05> 3<F3.821e+05> convexity (a relaxation of<F3.302e+05><F3.821e+05> mconvexity) which is closely related to semisimplicial orderings of graphs. We present new characterizations of HHDfree graphs via<F3.302e+05> m<F2.785e+05> 3<F3.821e+05> convexity and obtain some results known from [B. Jamison and S. Olariu, Adv. Appl. Math., 9 (1988), pp. 364376] as corollaries. Moreover, we characterize weak bipolarizable graphs as the graphs for which the family of all<F3.302e+05> m<F2.785e+05> 3<F3.821e+05> convex sets is a convex geometry. As an application of our results we present a simple efficient criterion for deciding whether a HHDfree graph contains a rdominating clique with respect to a given vertex radius function r.
Collective Additive Tree Spanners of Homogeneously Orderable Graphs (Extended Abstract)
, 2008
"... In this paper we investigate the (collective) tree spanners problem in homogeneously orderable graphs. This class of graphs was introduced by A. Brandstädt et al. to generalize the dually chordal graphs and the distancehereditary graphs and to show that the Steiner tree problem can still be solve ..."
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In this paper we investigate the (collective) tree spanners problem in homogeneously orderable graphs. This class of graphs was introduced by A. Brandstädt et al. to generalize the dually chordal graphs and the distancehereditary graphs and to show that the Steiner tree problem can still be solved in polynomial time on this more general class of graphs. In this paper, we demonstrate that every nvertex homogeneously orderable graph G admits – a spanning tree T such that, for any two vertices x, y of G, dT (x, y) ≤ dG(x, y) + 3 (i.e., an additive tree 3spanner) and – a system T (G) of at most O(log n) spanning trees such that, for any two vertices x, y of G, a spanning tree T ∈T(G) exists with dT (x, y) ≤ dG(x, y) + 2 (i.e, a system of at most O(log n) collective additive tree 2spanners). These results generalize known results on tree spanners of dually chordal graphs and of distancehereditary graphs. The results above are also complemented with some lower bounds which say that on some nvertex homogeneously orderable graphs any system of collective additive tree 1spanners must have at least Ω(n) spanning trees and there is no system of collective additive tree 2spanners with constant number of trees.
rDomination problems on trees and their homogeneous extensions
, 1995
"... . A graph is a homogeneous extension of a tree iff the reduction of all homogeneous sets (sometimes called modules) to single vertices gives a tree. We show that these graphs can be recognized in linear sequential and polylogarithmic parallel time using modular decomposition. As an application of so ..."
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. A graph is a homogeneous extension of a tree iff the reduction of all homogeneous sets (sometimes called modules) to single vertices gives a tree. We show that these graphs can be recognized in linear sequential and polylogarithmic parallel time using modular decomposition. As an application of some results on homogeneous sets we present a linear time algorithm computing the vertex sets of the connected components of the complement of an arbitrary graph. Moreover we present efficient parallel algorithms solving the problems rdominating set, r dominating clique and connected rdominating set (and thus the Steiner tree problem) on trees by reducing these problems to algebraic tree computations. Using these algorithms we can compute minimum rdominating cliques and minimum connected rdominating sets in homogeneous extensions of trees in linear sequential and logarithmic parallel time using a linear number of processors. Finally, we give a more involved sequential algorithm solv...