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Showing papers on "Dominating set published in 1992"


01 Jan 1992
TL;DR: It is shown that if Dominating Set is fixed-parameter tractable, then so are a variety of parameterized problems, such as Independent Set, and that for this problem, and for the problem of determining whether a graph has k disjoint cycles, it may take c = 1.
Abstract: For many fixed-parameter problems that are trivially solvable in polynomial-time, such as k-Dominating Set, essentially no better algorithm is presently known than the one which tries all possible solutions. Other problems, such as k-Feedback Vertex Set, exhibit fixed-parameter tractability: for each fixed k the problem is solvable in time bounded by a polynomial of degree c, where c is a constant independent of k. We show that for this problem, and for the problem of determining whether a graph has k disjoint cycles, we may take c = 1. We also show that if Dominating Set is fixed-parameter tractable, then so are a variety of parameterized problems, such as Independent Set. Some of the main results of a completeness theory for fixed-parameter intractability are surveyed.

309 citations


Journal ArticleDOI
TL;DR: It is proved that the MCBD case is NP-hard even when D = 2 and a polynomial heuristic for BCMD with a constant worst-case bound is described, which is no easier than finding such a heuristics for the dominating set problem.

80 citations


Journal ArticleDOI
TL;DR: It is NP-complete to decide (4) if there is a spanning tree with at least [exactly] n/2 + 1 leaves and (5) if G has a connected dominating set with cardinality ⩽n/2–1.

57 citations


Book ChapterDOI
19 Jun 1992
TL;DR: These approaches to polynomial time bounds are outlined and applied to two problems: minimum weight independent dominating set and maximum weight cycle-free subgraph (minimum weight feedback vertex set).
Abstract: For many problems on permutation graphs, polynomial time bounds were found by using different approaches as e.g. dynamic programming, structural properties of the intersection model, the reformulation as a shortest-path problem on suitable derived graphs and a geometric representation as points in the plane. Here we outline these approaches and apply them to two problems: minimum weight independent dominating set and maximum weight cycle-free subgraph (minimum weight feedback vertex set).

28 citations


Journal ArticleDOI
TL;DR: O(m +n log n) algorithms are presented for the minimum-weight connected dominating set and minimum- Weight Steiner subset of a permutation graph and if the graph is unweighted, a minimum-cardinality connected dominated set can be found in O(m) time.

19 citations


Journal ArticleDOI
TL;DR: This paper presents an optimal Θ(n log n) time algorithm for the problem of finding a minimum cardinality dominating cycle in a circular-arc graph.

11 citations


01 Jan 1992
TL;DR: A strong relationship is demonstrated between two application areas and the ideas of global domination and factoring and it is found that a factoring of a graph can represent the parallel computation of a class of constraint problems or the routes of multicast messages in a network.
Abstract: A factoring of a graph G = (V, E) is a collection of spanning subgraphs F$\sb1$,F$\sb2,$ ..., F$\sb{\rm k}$, known as factors into which the edge set E has been partitioned. A dominating set of a graph is a set of nodes such that every node in the graph is either contained in the set or has an edge to some node in the set. Each factor F$\sb{\rm i}$ is itself a graph and so has a dominating set. This set is called a local dominating sets or LDS. An LDS of minimum size contains $\gamma\sb{\rm i}$ nodes. In addition, there is some set of nodes named a global dominating set which dominates all of the factors. If a global dominating set is of minimum size, it is called a GDS and contains $\gamma$ nodes. A central question answered by this dissertation is under what circumstances, given a set of integers $\gamma\sb1,\gamma\sb2,\...,\gamma\sb{\rm k}$, and $\gamma$, there is a graph which can be factored into k factors in such a way that a minimum LDS of F$\sb{\rm i}$ has size $\gamma\sb{\rm i},$ 1 $\le$ i $\le$ k, and a GDS has size $\gamma$. The general solution to this central question is complicated. In addition, simpler subproblems are often precisely those which are most applicable to practical problems. For these reasons, simpler solutions are found for several special cases of the general characterization problem. A strong relationship is demonstrated between two application areas and the ideas of global domination and factoring. We find that a factoring of a graph can represent the parallel computation of a class of constraint problems or the routes of multicast messages in a network. The applicability of these ideas is limited by the computational complexity of the problem of finding a GDS in a factoring. The problem is NP-Hard in general and we find that, more surprisingly, when the factors are very simple structures such as trees or even paths, the problem remains NP-Hard.

5 citations


Journal ArticleDOI
01 Jul 1992
TL;DR: A novel bipartite graph theoretic approach to spare allocation is presented for structures with dedicated spares and direct replacement reconfiguration and is shown to be equivalent to either a graph matching or a graph dominating set problem.
Abstract: One approach to enhancing the yield or reliability of large area VLSI structures has been by means of spare interconnect, logic and computational units Although extensive literature exists concerning design for inclusion of spares and restructuring mechanisms in memories and processor arrays, little research has been published on general models and algorithms for broad classes of spare allocation and reconfiguration problems The main contribution of the paper is that a novel bipartite graph theoretic approach to spare allocation is presented for structures with dedicated spares and direct replacement reconfiguration Spare allocation for classes of reconfigurable structures is shown to be equivalent to either a graph matching or a graph dominating set problem The complexity of optimal spare allocation for each of the problem classes is described, and reconfiguration algorithms are provided Several detailed examples are presented, and implementation of the algorithms is discussed

4 citations


Journal Article
TL;DR: In this article, it was shown that every longest cycle in a 2-connected graph is not a dominating cycle unless G is a spanning subgraph of a graph belonging to one of four easily specified classes of graphs.
Abstract: Let G be a 2-connected graph on n vertices such that d(x) + d(y) + d(z) n for all triples of independent vertices x, y, z. We prove that every longest cycle in G is a dominating cycle unless G is a spanning subgraph of a graph belonging to one of four easily specified classes of graphs.

3 citations


Journal ArticleDOI
TL;DR: A new linear algorithm for the minimum weight total dominating set problem for trees, given a new proof of this fact by directly constructing a strong elimination ordering.

1 citations


Proceedings ArticleDOI
01 Mar 1992
TL;DR: In WIS paper, NC algorithms for finding minimum weight independent dominating set, minimum weight total dominating set (MWTDS), and minimum weight connected dominating set for pemnutation grapha take O (log 2n) time on CREW PRAM.
Abstract: In WIS paper, we present NC algorithms for finding minimum weight independent dominating set (MWIDS), minimum weight total dominating set (MWTDS), and minimum weight connected dominating set (MWCDS) for pemnutation grapha. All these algorithms take O (log 2n) time on CREW PRAM. The number of processors required is n3/logn for finding MWIDS, and mWogrn for tinding MWTDS and MWCDS, wheze m is the number of edges in a permutation graph of n nodes.

Book ChapterDOI
TL;DR: In this paper, the authors present some numerical invariants of graphs that are related to the concept of domination, namely, the domatic number and its variants, which is the maximum number of classes of a partition of V (G) into dominating sets that induce connected subgraphs of G.
Abstract: Publisher Summary This chapter presents some numerical invariants of graphs that are related to the concept of domination—namely, the domatic number and its variants.. The word domatic was coined from the words dominating and chromatic in the same way as the word smog was composed from the words smoke and fog. This concept is a certain analogy of the chromatic number, but instead of independent sets, dominating sets are used in its definition. A subset D of the vertex set V ( G ) of an undirected graphs G is called dominating if for each x V ( G ) − D there exists a vertex y D adjacent to x . A domatic partition of G is a partition of V ( G ), all of whose classes are dominating sets in G . The maximum number of classes of a domatic partition of G is called the “domatic number” of G and denoted by d ( G ). R. Laskar and S. T. Hedetniemi have introduced the connected domatic number d , ( G ) of a graph G . It is the maximum number of classes of a partition of V ( G ) into dominating sets that induce connected subgraphs of G .

30 Oct 1992
TL;DR: It is shown that a maximal co-hitting set for S can be computed on an EREW PRAM in time O(np(log(n + m))2) using 0 ( n 2 m ) processors, which implies that if ap = 0 ((log( n f m))" then the problem is solvable in N C.
Abstract: Let C = {el,. . . , c,) be a family of subsets of a finite set S = (1,. . . , n), a subset S' of S is a co-hitting set if S' contains no element of C as a subset. By using an O((10gn)~) time EREW PRAM algorithm for a maximal independent set problem (MIS), we show that a maximal co-hitting set for S can be computed on an EREW PRAM in time O(np(log(n + m))2) using 0 ( n 2 m ) processors, where a = max{lc;l I i = 1,. . . , rn) and /? = max{ldjl I j = 1,. . . , n ) with dj = {c; I j E ci) This implies that if ap = 0 ((log(n f m))" then the problem is solvable in N C. I