共查询到20条相似文献,搜索用时 31 毫秒
1.
Toru Araki 《Information Processing Letters》2007,104(3):86-90
In a digraph G, a vertex u is said to dominate itself and vertices v such that (u,v) is an arc of G. For a positive integer k, a k-tuple dominating set D of a digraph is a subset of vertices such that every vertex is dominated by at least k vertices in D. The k-tuple domination number of a given digraph is the minimum cardinality of a k-tuple dominating set of the digraph. In this letter, we give the exact values of the k-tuple domination number of de Bruijn and Kautz digraphs. 相似文献
2.
Let G=(V,A) be a digraph. A set T of vertices of G is a twin dominating set of G if for every vertex v∈V?T, there exist u,w∈T (possibly u=w) such that arcs (u,v),(v,w)∈A. The twin domination numberγ∗(G) of G is the cardinality of a minimum twin dominating set of G. In this paper we investigate the twin domination number in generalized de Bruijn digraphs GB(n,d). For the digraphs GB(n,d), we first establish sharp bounds on the twin domination number. Secondly, we give the exact values of the twin domination number for several types of GB(n,d) by constructing minimum twin dominating sets in the digraphs. Finally, we present sharp upper bounds for some special generalized de Bruijn digraphs. 相似文献
3.
《国际计算机数学杂志》2012,89(3):522-526
A set S of vertices of a graph G is a dominating set for G if every vertex of G is adjacent to at least one vertex of S. The domination number γ(G), of G, is the minimum cardinality of a dominating set in G. Moreover, if the maximum degree of G is Δ, then for every positive integer k≤Δ, the set S is a k-dominating set in G if every vertex outside of S is adjacent to at least k vertices of S. The k-domination number of G, denoted by γ k (G), is the minimum cardinality of a k-dominating set in G. A map f: V→<texlscub>0, 1, 2</texlscub>is a Roman dominating function for G if for every vertex v with f(v)=0, there exists a vertex u∈N(v) such that f(u)=2. The weight of a Roman dominating function is f(V)=∑ u∈V f(u). The Roman domination number γR(G), of G, is the minimum weight of a Roman dominating function on G. In this paper, we obtain that for any two graphs G and H, the k-domination number of the Cartesian product of G and H is bounded below by γ(G)γ k (H)/2. Also, we obtain that the domination number of Cartesian product of G and H is bounded below by γ(G)γR(H)/3. 相似文献
4.
k-tuple domination in graphs 总被引:1,自引:0,他引:1
Chung-Shou Liao 《Information Processing Letters》2003,87(1):45-50
In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the k-tuple domination problem is to find a minimum sized vertex subset in a graph such that every vertex in the graph is dominated by at least k vertices in this set. The current paper studies k-tuple domination in graphs from an algorithmic point of view. In particular, we give a linear-time algorithm for the k-tuple domination problem in strongly chordal graphs, which is a subclass of chordal graphs and includes trees, block graphs, interval graphs and directed path graphs. We also prove that the k-tuple domination problem is NP-complete for split graphs (a subclass of chordal graphs) and for bipartite graphs. 相似文献
5.
《Journal of Parallel and Distributed Computing》2001,61(8):1013-1032
We propose broadcasting algorithms for line digraphs in the telegraph model. The new protocols use a broadcasting protocol for a graph G to obtain a broadcasting protocol for the graph LkG, the graph obtained by applying k times, the line digraph operation to G. As a consequence improved bounds for the broadcasting time in De Bruijn, Kautz, and Wrapped Butterfly digraphs are obtained. 相似文献
6.
In this paper, we focus on the oriented coloring of graphs. Oriented coloring is a coloring of the vertices of an oriented graph G without symmetric arcs such that (i) no two neighbors in G are assigned the same color, and (ii) if two vertices u and v such that (u,v)∈A(G) are assigned colors c(u) and c(v), then for any (z,t)∈A(G), we cannot have simultaneously c(z)=c(v) and c(t)=c(u). The oriented chromatic number of an unoriented graph G is the smallest number k of colors for which any of the orientations of G can be colored with k colors.The main results we obtain in this paper are bounds on the oriented chromatic number of particular families of planar graphs, namely 2-dimensional grids, fat trees and fat fat trees. 相似文献
7.
Given a graph G, the problem is to construct a smallest subset S of vertices whose deletion results in an acyclic subgraph. The set S is called a minimum feedback vertex set for G.Tight upper and lower bounds on the cardinality of minimum feedback vertex sets have been previously obtained for some hypercube-like networks, such as meshes, tori, butterflies, cube-connected cycles and hypercubes. In this paper we construct minimum feedback vertex sets and determine their cardinalities in certain shuffle-based interconnection networks, such as shuffle-exchange, de Bruijn and Kautz networks. 相似文献
8.
Ruijuan Li 《Information Processing Letters》2010,110(16):651-654
For a positive integer k, a graph G is k-ordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. In this paper, we show that if G is a ⌊3k/2⌋-connected graph of order n?100k, and d(u)+d(v)?n for any two vertices u and v with d(u,v)=2, then G is k-ordered hamiltonian. Our result implies the theorem of G. Chen et al. [Ars Combin. 70 (2004) 245-255] [1], which requires the degree sum condition for all pairs of non-adjacent vertices, not just those distance 2 apart. 相似文献
9.
Meijie Ma 《Information Sciences》2010,180(17):3373-3379
A k-container of a graph G is a set of k internally disjoint paths between u and v. A k-container of G is a k∗-container if it contains all vertices of G. A graph G is k∗-connected if there exists a k∗-container between any two distinct vertices, and a bipartite graph G is k∗-laceable if there exists a k∗-container between any two vertices u and v from different partite sets of G for a given k. A k-connected graph (respectively, bipartite graph) G is f-edge fault-tolerant spanning connected (respectively, laceable) if G − F is w∗-connected for any w with 1 ? w ? k − f and for any set F of f faulty edges in G. This paper shows that the folded hypercube FQn is f-edge fault-tolerant spanning laceable if n(?3) is odd and f ? n − 1, and f-edge fault-tolerant spanning connected if n (?2) is even and f ? n − 2. 相似文献
10.
《国际计算机数学杂志》2012,89(4):584-588
A k-dominating set for a graph G(V, E) is a set of vertices D? V such that every vertex v∈V\ D is adjacent to at least k vertices in D. The k-domination number of G, denoted by γ k (G), is the cardinality of a smallest k-dominating set of G. Here we establish lower and upper bounds of γ k (C m ×C n ) for k=2. In some cases, these bounds agree so that the exact 2-domination number is obtained. 相似文献
11.
Suppose the vertices of a graph G were labeled arbitrarily by positive integers, and let S(v) denote the sum of labels over all neighbors of vertex v. A labeling is lucky if the function S is a proper coloring of G, that is, if we have S(u)≠S(v) whenever u and v are adjacent. The least integer k for which a graph G has a lucky labeling from the set {1,2,…,k} is the lucky number of G, denoted by η(G).Using algebraic methods we prove that η(G)?k+1 for every bipartite graph G whose edges can be oriented so that the maximum out-degree of a vertex is at most k. In particular, we get that η(T)?2 for every tree T, and η(G)?3 for every bipartite planar graph G. By another technique we get a bound for the lucky number in terms of the acyclic chromatic number. This gives in particular that for every planar graph G. Nevertheless we offer a provocative conjecture that η(G)?χ(G) for every graph G. 相似文献
12.
In a graph G, a k-container Ck(u,v) is a set of k disjoint paths joining u and v. A k-container Ck(u,v) is k∗-container if every vertex of G is passed by some path in Ck(u,v). A graph G is k∗-connected if there exists a k∗-container between any two vertices. An m-regular graph G is super-connected if G is k∗-connected for any k with 1?k?m. In this paper, we prove that the recursive circulant graphs G(2m,4), proposed by Park and Chwa [Theoret. Comput. Sci. 244 (2000) 35-62], are super-connected if and only if m≠2. 相似文献
13.
The k-ary n-cube has been one of the most popular interconnection networks for massively parallel systems. Given a set P of at most 2n − 2 (n ? 2) prescribed edges and two vertices u and v, we show that the 3-ary n-cube contains a Hamiltonian path between u and v passing through all edges of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths, none of them having u or v as internal vertices or both of them as end-vertices. As an immediate result, the 3-ary n-cube contains a Hamiltonian cycle passing through a set P of at most 2n − 1 prescribed edges if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths. 相似文献
14.
A k-containerC(u,v) of a graph G is a set of k disjoint paths joining u to v. A k-container C(u,v) is a k∗-container if every vertex of G is incident with a path in C(u,v). A bipartite graph G is k∗-laceable if there exists a k∗-container between any two vertices u, v from different partite set of G. A bipartite graph G with connectivity k is super laceable if it is i∗-laceable for all i?k. A bipartite graph G with connectivity k is f-edge fault-tolerant super laceable if G−F is i∗-laceable for any 1?i?k−f and for any edge subset F with |F|=f<k−1. In this paper, we prove that the hypercube graph Qr is super laceable. Moreover, Qr is f-edge fault-tolerant super laceable for any f?r−2. 相似文献
15.
In this paper, we initiate the study of a variation of standard domination, namely total restrained domination. Let G=(V,E) be a graph. A set D⊆V is a total restrained dominating set if every vertex in V−D has at least one neighbor in D and at least one neighbor in V−D, and every vertex in D has at least one neighbor in D. The total restrained domination number of G, denoted by γtr(G), is the minimum cardinality of all total restrained dominating sets of G. We determine the best possible upper and lower bounds for γtr(G), characterize those graphs achieving these bounds and find the best possible lower bounds for where both G and are connected. 相似文献
16.
In this paper, we investigate isomorphic factorizations of the Kronecker product graphs. Using these relations, it is shown that (1) the Kronecker product of the d-out-regular digraph and the complete symmetric digraph is factorized into the line digraph, (2) the Kronecker product of the Kautz digraph and the de Bruijn digraph is factorized into the Kautz digraph, (3) the Kronecker product of binary generalized de Bruijn digraphs is factorized into the binary generalized de Bruijn digraph. 相似文献
17.
Let f be a function on pairs of vertices. An f
-labeling scheme for a family of graphs ℱ labels the vertices of all graphs in ℱ such that for every graph G∈ℱ and every two vertices u,v∈G, f(u,v) can be inferred by merely inspecting the labels of u and v. The size of a labeling scheme is the maximum number of bits used in a label of any vertex in any graph in ℱ. This paper illustrates
that the notion of universal matrices can be used to efficiently construct f-labeling schemes. 相似文献
18.
Xiuying Li 《Information Processing Letters》2010,110(22):986-991
For a graph G=(V,E), a subset D⊆V is an r-hop dominating set if every vertex u∈V−D is at most r-hops away from D. It is a 2-connected r-hop dominating set if the subgraph of G induced by D is 2-connected. In this paper, we present two approximation algorithms to compute minimum 2-connected r-hop dominating set. The first one is a greedy algorithm using ear decomposition of 2-connected graphs. This algorithm is applicable to any 2-connected general graph. The second one is a three-phase algorithm which is only applicable to unit disk graphs. For both algorithms, performance ratios are given. 相似文献
19.
《国际计算机数学杂志》2012,89(5):912-920
A vertex v of a connected graph G distinguishes a pair u, w of vertices of G if d(v, u)≠d(v, w), where d(·,·) denotes the length of a shortest path between two vertices in G. A k-partition Π={S 1, S 2, …, S k } of the vertex set of G is said to be a locatic partition if for every pair of distinct vertices v and w of G, there exists a vertex s∈S i for all 1≤i≤k that distinguishes v and w. The cardinality of a largest locatic partition is called the locatic number of G. In this paper, we study the locatic number of paths, cycles and characterize all the connected graphs of order n having locatic number n, n?1 and n?2. Some realizable results are also given in this paper. 相似文献
20.
Let λ(G) be the edge connectivity of G. The direct product of graphs G and H is the graph with vertex set V(G×H)=V(G)×V(H), where two vertices (u1,v1) and (u2,v2) are adjacent in G×H if u1u2∈E(G) and v1v2∈E(H). We prove that λ(G×Kn)=min{n(n−1)λ(G),(n−1)δ(G)} for every nontrivial graph G and n?3. We also prove that for almost every pair of graphs G and H with n vertices and edge probability p, G×H is k-connected, where k=O(2(n/logn)). 相似文献