Showing papers on "Dominating set published in 1995"

Fixed-Parameter Tractability and Completeness I: Basic Results

Abstract: For many fixed-parameter problems that are trivially soluable 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 soluable in time bounded by a polynomial of degree $c$, where $c$ is a constant independent of $k$. We establish the main results of a completeness program which addresses the apparent fixed-parameter intractability of many parameterized problems. In particular, we define a hierarchy of classes of parameterized problems $FPT \subseteq W[1] \subseteq W[2] \subseteq \cdots \subseteq W[SAT] \subseteq W[P]$ and identify natural complete problems for $W[t]$ for $t \geq 2$. (In other papers we have shown many problems complete for $W[1]$.) DOMINATING SET is shown to be complete for $W[2]$, and thus is not fixed-parameter tractable unless INDEPENDENT SET, CLIQUE, IRREDUNDANT SET and many other natural problems in $W[2]$ are also fixed-parameter tractable. We also give a compendium of currently known hardness results as an appendix.

Fast distributed construction of k-dominating sets and applications

Abstract: This paper presents a fast distributed algorithm to compute a small k-dominating set D (for any fixed k) and its induced graph partition (breaking the graph into radius k clusters centered around the vertices of D). The time complexity of the algorithm is O(k log* n). Small k-dominating sets have applications in a number of areas, including routing with sparse routing tables via the scheme of [P~, the design of distributed data structures [P2], and center selection in a distributed network (cf. [BKP]). The main application described in this paper concerns a fast distributed algorithm for constructing a minimum weight spanning tree (MST). On an n-vertex network of diameter d, the new algorithm constructs an MST in time O(@log* n + d), improving on the results of [GKP]. The new MST algorithm is conceptually simpler than the three-phase algorithm of [GKP]. In addition to exploiting small k-dominating sets, it uses a very simple convergecast protocol to inform a center about graph edges, that avoids forwarding messages about edges that close cycles. This convergecast protocol is similar to the one used in the third phase of the algorithm of [GKP], and most of the novelty lies in a new careful analysis proving that the convergecast process is fully pipelined, and no waiting occurs at intermediate nodes. This enables the new algorithm to skip the complicated second phase of the algorithm of [GKP].

An induced subgraph characterization of domination perfect graphs

Abstract: Let γ(G) ι(G) be the domination number and independent domination number of a graph (G), respectively. A graph (G) is called domination perfect if γ(H) = ι(H), for every induced subgraph H of (G). There are many results giving a partial characterization of domination perfect graphs. In this paper, we present a finite induced subgraph characterization of the entire class of domination perfect graphs. The list of forbidden subgraphs in the charcterization consists of 17 minimal domination imperfect graphs. Moreover, the dominating set and independent dominating set problems are shown to be both NP‐complete on some classes of graphs. © 1995 John Wiley & Sons, Inc. Copyright © 1995 Wiley Periodicals, Inc., A Wiley Company

Diametral Path Graphs

Abstract: We introduce a new class of graphs called diametral path graphs that properly contains the class of asteroidal triple-free graphs and the class of dominating pair graphs. We characterize the trees as well as the chordal graphs that are diametral path graphs. We present an O(n3m) algorithm deciding whether a given graph has a dominating diametral path. Finally, we study the structure of minimum connected dominating sets in diametral path graphs.

The edge domination problem

Abstract: An edge dominating set of a graph is a set D of edges such that every edge not in D is adjacent to at least one edge in D: In this paper we present a linear time algorithm for flnding a minimum edge dominating set of a block graph.

Linear Time Algorithms for Dominating Pairs in Asteroidal Triple-free Graphs

Abstract: 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 triple-free (AT-free, for short) if it contains no asteroidal triple. The motivation for this investigation is provided, in part, by the fact that AT-free 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 AT-free 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 AT-free graphs. The resulting simple algorithm, based on the well-known Lexicographic Breadth-First Search, 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(¦V¦3) for input graph G=(V, E). In addition, we indicate how our algorithm can be extended to find, in time linear in the size of the input, all dominating pairs in a connected AT-free graph with diameter greater than three. A remarkable feature of the extended algorithm is that, even though there may be O(¦V¦2) dominating pairs, the algorithm can compute and represent them in linear time.

Computing a Dominating Pair in an Asteroidal Triple-free Graph in Linear Time

Abstract: 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 triple-free (AT-free, for short) if it contains no asteroidal triple. The motivation for this work is provided, in part, by the fact that AT-free 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 AT-free 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 AT-free 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(¦V¦3) for input graph G=(V, E).

Convexity of Minimal Total Dominating Functions in Graphs

Abstract: A total dominating function (TDF) of a graph G=(V, E) is a function f ∶ V → [0,1] such that for each v ∈ V, the sum of f values over all neighbours of ν (i.e., all vertices adjacent to v) is at least one. Integer-valued TDFs are precisely the characteristic functions of total dominating sets of G. A minimal TDF (MTDF) is one such that decreasing any value of it makes it non-TDF. An MTDF f is called universal if convex combinations of f and any other MTDF are minimal. We give a sufficient condition for an MTDF to be universal which generalizes previous results. Also we define a splitting operation on a graph G as follows: take any vertex ν in G and a vertex ω not in G and join ω with all the neighbours of v. A graph G has a universal MTDF if and only if the graph obtained by splitting G has a universal MTDF. A corollary is that graphs obtained by the operation from paths, cycles, complete graphs, wheels, and caterpillar graphs have a universal MTDF.

Highly parallelizable problems on sorted intervals

Abstract: We show that several combinatorial optimization problems on an interval graph given its interval representation in sorted order, are highly parallelizable in the sense of Berkman et al [2]. For each of these problems, we present an O(log log n) time parallel algorithm which uses O( n log log n ) processors on the Common CRCW PRAM model which is the weakest of among the CRCW PRAM models. Our algorithms are optimal since, all the problems under consideration can be solved sequentially in O(n) time given a sorted interval set. The problems we solve are finding minimum dominating set (MDS), minimum connected dominating set (MCDS), minimum total dominating set (MTDS), maximum independent set (MIS), and depth first search tree (DFS) and breadth first search tree (BFS) starting from a vertex corresponding to an arbitrary interval. No previous highly parallelizable combinatorial problems either on graph structures or on its geometrical representations are known. Optimal parallel algorithms with O(log n) running time are known for MDS, DFS, BFS, and MIS problems.

Zero-Knowledge Proofs for Independent Set and Dominating Set Problems

Abstract: Independent set and dominating set problems are two NP-C problems in graph theory. In this paper we propose two zero-knowledge proofs for these problems.