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1.
The most popular bounded-degree derivative network of the hypercube is the butterfly network. The Benes network consists of back-to-back butterflies. There exist a number of topological representations that are used to describe butterfly—like architectures. We identify a new topological representation of butterfly and Benes networks.The minimum metric dimension problem is to find a minimum set of vertices of a graph G(V,E) such that for every pair of vertices u and v of G, there exists a vertex w with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. It is NP-hard in the general sense. We show that it remains NP-hard for bipartite graphs. The algorithmic complexity status of this NP-hard problem is not known for butterfly and Benes networks, which are subclasses of bipartite graphs. By using the proposed new representations, we solve the minimum metric dimension problem for butterfly and Benes networks. The minimum metric dimension problem is important in areas such as robot navigation in space applications.  相似文献   

2.
Let G be a connected (di)graph. A vertex w is said to strongly resolve a pair u,v of vertices of G if there exists some shortest u-w path containing v or some shortest v-w path containing u. A set W of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of W. The smallest cardinality of a strong resolving set for G is called the strong dimension of G. It is shown that the problem of finding the strong dimension of a connected graph can be transformed to the problem of finding the vertex covering number of a graph. Moreover, it is shown that computing this invariant is NP-hard. Related invariants for directed graphs are defined and studied.  相似文献   

3.
Let G be a graph. For u,vV(G) with distG(u,v)=2, denote JG(u,v)={wNG(u)∩NG(v)|NG(w)NG(u)NG(v){u,v}}. A graph G is called quasi claw-free if JG(u,v)≠ for any u,vV(G) with distG(u,v)=2. In 1986, Thomassen conjectured that every 4-connected line graph is hamiltonian. In this paper we show that every 4-connected line graph of a quasi claw-free graph is hamiltonian connected.  相似文献   

4.
Given a graph G = (V, E), a set W í V{W \subseteq V} is said to be a resolving set if for each pair of distinct vertices u, v ? V{u, v \in V} there is a vertex x in W such that d(u, x) 1 d(v, x){d(u, x) \neq d(v, x)} . The resolving number of G is the minimum cardinality of all resolving sets. In this paper, conditions are imposed on resolving sets and certain conditional resolving parameters are studied for honeycomb and hexagonal networks.  相似文献   

5.
We investigate the following modification of the well-known irregularity strength of graphs. Given a total weighting w of a graph G=(V,E) with elements of a set {1,2,…,s}, denote wtG(v)=∑evw(e)+w(v) for each vV. The smallest s for which exists such a weighting with wtG(u)≠wtG(v) whenever u and v are distinct vertices of G is called the total vertex irregularity strength of this graph, and is denoted by . We prove that for each graph of order n and with minimum degree δ>0.  相似文献   

6.
The concept of metric basis is useful for robot navigation. In graph G, a robot is aware of its current location by sending signals to obtain the distances between itself and the landmarks in G. Its position is determined uniquely in G if it knows its distances to sufficiently many landmarks. The metric basis of G is defined as the minimum set of landmarks such that all other vertices in G can be uniquely determined and the metric dimension of G is defined as the cardinality of the minimum set of landmarks. The major contribution of this paper is that we have partly solved the open problem proposed by Manuel et al. [9], by proving that the metric dimension of HDN1(n) and HDN2(n) are either 3 or 4. However, the problem of finding the exact metric dimension of HDN networks is still open.  相似文献   

7.
Lan Xu  Baoyindureng Wu   《Discrete Mathematics》2008,308(22):5144-5148
The transformation graph G-+- of a graph G is the graph with vertex set V(G)E(G), in which two vertices u and v are joined by an edge if one of the following conditions holds: (i) u,vV(G) and they are not adjacent in G, (ii) u,vE(G) and they are adjacent in G, (iii) one of u and v is in V(G) while the other is in E(G), and they are not incident in G. In this paper, for any graph G, we determine the connectivity and the independence number of G-+-. Furthermore, for a graph G of order n4, we show that G-+- is hamiltonian if and only if G is not isomorphic to any graph in {2K1+K2,K1+K3}{K1,n-1,K1,n-1+e,K1,n-2+K1}.  相似文献   

8.
Let G be a connected, undirected graph without loops and without multiple edges. For a pair of distinct vertices u and v, a minimum {u, v}-separating set is a smallest set of edges in G whose removal disconnects u and v. The edge connectivity of G, denoted λ(G), is defined to be the minimum cardinality of a minimum {u, v}-separating set as u and v range over all pairs of distinct vertices in G. We introduce and investigate the eavesdropping number, denoted ε(G), which is defined to be the maximum cardinality of a minimum {u, v}-separating set as u and v range over all pairs of distinct vertices in G. Results are presented for regular graphs and maximally locally connected graphs, as well as for a number of common families of graphs.  相似文献   

9.
The distancedG(u,v) between two vertices u and v in a connected graph G is the length of the shortest (u,v) path in G. A (u,v) path of length dG(u,v) is called a (u,v)-geodesic. A set XV is called weakly convex in G if for every two vertices a,bX, exists an (a,b)-geodesic, all of whose vertices belong to X. A set X is convex in G if for all a,bX all vertices from every (a,b)-geodesic belong to X. The weakly convex domination number of a graph G is the minimum cardinality of a weakly convex dominating set of G, while the convex domination number of a graph G is the minimum cardinality of a convex dominating set of G. In this paper we consider weakly convex and convex domination numbers of tori.  相似文献   

10.
For an ordered set W = {w 1, w 2,..., w k} of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the k-vector r(v|W) = (d(v, w 1), d(v, w 2),... d(v, w k)), where d(x, y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct representations with respect to W. A resolving set for G containing a minimum number of vertices is a basis for G. The dimension dim(G) is the number of vertices in a basis for G. A resolving set W of G is connected if the subgraph 〈W〉 induced by W is a nontrivial connected subgraph of G. The minimum cardinality of a connected resolving set in a graph G is its connected resolving number cr(G). Thus 1 ≤ dim(G) ≤ cr(G) ≤ n?1 for every connected graph G of order n ≥ 3. The connected resolving numbers of some well-known graphs are determined. It is shown that if G is a connected graph of order n ≥ 3, then cr(G) = n?1 if and only if G = K n or G = K 1,n?1. It is also shown that for positive integers a, b with ab, there exists a connected graph G with dim(G) = a and cr(G) = b if and only if $\left( {a,b} \right) \notin \left\{ {\left( {1,k} \right):k = 1\;{\text{or}}\;k \geqslant 3} \right\}$ Several other realization results are present. The connected resolving numbers of the Cartesian products G × K 2 for connected graphs G are studied.  相似文献   

11.
The Randi? indexR(G) of a graph G is defined as the sum of over all edges uv of G, where du and dv are the degrees of vertices u and v, respectively. Let D(G) be the diameter of G when G is connected. Aouchiche et al. (2007) [1] conjectured that among all connected graphs G on n vertices the path Pn achieves the minimum values for both R(G)/D(G) and R(G)−D(G). We prove this conjecture completely. In fact, we prove a stronger theorem: If G is a connected graph, then , with equality if and only if G is a path with at least three vertices.  相似文献   

12.
For two vertices u and v of a connected graph G, the set I(u,v) consists of all those vertices lying on a u-v geodesic in G. For a set S of vertices of G, the union of all sets I(u,v) for u, v S is denoted by I(S). A set S is a convex set if I(S) = S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. A convex set S in G with |S| = con(G) is called a maximum convex set. A subset T of a maximum convex set S of a connected graph G is called a forcing subset for S if S is the unique maximum convex set containing T. The forcing convexity number f(S, con) of S is the minimum cardinality among the forcing subsets for S, and the forcing convexity number f(G, con) of G is the minimum forcing convexity number among all maximum convex sets of G. The forcing convexity numbers of several classes of graphs are presented, including complete bipartite graphs, trees, and cycles. For every graph G, f(G, con) con(G). It is shown that every pair a, b of integers with 0 a b and b is realizable as the forcing convexity number and convexity number, respectively, of some connected graph. The forcing convexity number of the Cartesian product of H × K 2 for a nontrivial connected graph H is studied.  相似文献   

13.
A hamiltonian cycle C of a graph G is an ordered set u1,u2,…,un(G),u1 of vertices such that uiuj for ij and ui is adjacent to ui+1 for every i{1,2,…,n(G)−1} and un(G) is adjacent to u1, where n(G) is the order of G. The vertex u1 is the starting vertex and ui is the ith vertex of C. Two hamiltonian cycles C1=u1,u2,…,un(G),u1 and C2=v1,v2,…,vn(G),v1 of G are independent if u1=v1 and uivi for every i{2,3,…,n(G)}. A set of hamiltonian cycles {C1,C2,…,Ck} of G is mutually independent if its elements are pairwise independent. The mutually independent hamiltonicity IHC(G) of a graph G is the maximum integer k such that for any vertex u of G there exist k mutually independent hamiltonian cycles of G starting at u.In this paper, the mutually independent hamiltonicity is considered for two families of Cayley graphs, the n-dimensional pancake graphs Pn and the n-dimensional star graphs Sn. It is proven that IHC(P3)=1, IHC(Pn)=n−1 if n≥4, IHC(Sn)=n−2 if n{3,4} and IHC(Sn)=n−1 if n≥5.  相似文献   

14.
All-Pairs Small-Stretch Paths   总被引:1,自引:0,他引:1  
Let G = (VE) be a weighted undirected graph. A path between uv  V is said to be of stretch t if its length is at most t times the distance between u and v in the graph. We consider the problem of finding small-stretch paths between all pairs of vertices in the graph G.It is easy to see that finding paths of stretch less than 2 between all pairs of vertices in an undirected graph with n vertices is at least as hard as the Boolean multiplication of two n × n matrices. We describe three algorithms for finding small-stretch paths between all pairs of vertices in a weighted graph with n vertices and m edges. The first algorithm, STRETCH2, runs in Õ(n3/2m1/2) time and finds stretch 2 paths. The second algorithm, STRETCH7/3, runs in Õ(n7/3) time and finds stretch 7/3 paths. Finally, the third algorithm, STRETCH3, runs in Õ(n2) and finds stretch 3 paths.Our algorithms are simpler, more efficient and more accurate than the previously best algorithms for finding small-stretch paths. Unlike all previous algorithms, our algorithms are not based on the construction of sparse spanners or sparse neighborhood covers.  相似文献   

15.
The distance between a pair of vertices u, v in a graph G is the length of a shortest path joining u and v. The diameter diam(G) of G is the maximum distance between all pairs of vertices in G. A spanning tree T of G is diameter preserving if diam(T) = diam(G). In this note, we characterize graphs that have diameter-preserving spanning trees.  相似文献   

16.
17.
Let G be a graph and v be any vertex of G. Then the neighborhood contracted graphGv of G, with respect to the vertex v, is the graph with vertex set V ? N(v), where two vertices u,wV ? N(v) are adjacent in Gv if either w = v and u is adjacent to any vertex of N(v) in G or u,w ? N[v] and u,w are adjacent in G. The properties of the neighborhood contracted graphs are discussed in this paper. The neighborhood contraction in some special class of graphs, the domination in a graph and the neighborhood contracted graphs are discussed in the paper.  相似文献   

18.
The n-cube is characterized as a connected regular graph in which for any three vertices u, v, and w there is a unique vertex that lies simultaneously on a shortest (u, v)-path, a shortest (v, w)-path, and a shortest (w, u)-path.  相似文献   

19.
For a vertex v of a graph G, we denote by d(v) the degree of v. The local connectivity κ(u, v) of two vertices u and v in a graph G is the maximum number of internally disjoint uv paths in G, and the connectivity of G is defined as κ(G)=min{κ(u, v)|u, vV(G)}. Clearly, κ(u, v)?min{d(u), d(v)} for all pairs u and v of vertices in G. Let δ(G) be the minimum degree of G. We call a graph G maximally connected when κ(G)=δ(G) and maximally local connected when for all pairs u and v of distinct vertices in G. In 2006, Hellwig and Volkmann (J Graph Theory 52 (2006), 7–14) proved that a connected graph G with given clique number ω(G)?p of order n(G) is maximally connected when As an extension of this result, we will show in this work that these conditions even guarantee that G is maximally local connected. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 192–197, 2010  相似文献   

20.
Graph Connectivity After Path Removal   总被引:1,自引:0,他引:1  
Let G be a graph and u, v be two distinct vertices of G. A u—v path P is called nonseparating if G—V(P) is connected. The purpose of this paper is to study the number of nonseparating u—v path for two arbitrary vertices u and v of a given graph. For a positive integer k, we will show that there is a minimum integer (k) so that if G is an (k)-connected graph and u and v are two arbitrary vertices in G, then there exist k vertex disjoint paths P 1[u,v], P 2[u,v], . . ., P k [u,v], such that G—V (P i [u,v]) is connected for every i (i = 1, 2, ..., k). In fact, we will prove that (k) 22k+2. It is known that (1) = 3.. A result of Tutte showed that (2) = 3. We show that (3) = 6. In addition, we prove that if G is a 5-connected graph, then for every pair of vertices u and v there exists a path P[u, v] such that G—V(P[u, v]) is 2-connected.* Supported by NSF grant No. DMS-0070059 Supported by ONR grant N00014-97-1-0499 Supported by NSF grant No. 9531824  相似文献   

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