Accurate Connected Edge Domination Number in Graphs

Let γPm □ Cn denote the domination number of the cylindrical grid graph formed by the Cartesian product of the graphs Pm, the path of length m, m≥2, and the graph Cn, the cycle of length n, n≥3. In this paper we propose methods to find the domination numbers of graphs of the form Pm □ Cn with n≥3 and m=5 and propose tight bounds on domination numbers of the graphs P6 □ Cn, n≥3. Moreover, we provide rough bounds on domination numbers of the graphs Pm □ Cn, n≥3 and m≥7. We also point out how domination numbers and minimum dominating sets are useful for wireless sensor networks.

A edge dominating set F of a graph G = (V , E) is an accurate edge dominating set, if < E  F > has no edge dominating set of cardinality | F | . The accurate edge domination number (G) ae  is the minimum cardinality of an accurate edge dominating set. We study the graph theoretic properties of (G) ae  and its exact values for some standard graphs. The relation between (G) ae  with other parameters are also investigated

A subset $X$ of edges of a graph $G$ is called an \textit{edge dominating set} of $G$ if every edge not in $X$ is adjacent to some edge in $X$. The edge domination number $\gamma'(G)$ of $G$ is the minimum cardinality taken over all edge dominating sets of $G$. An \textit{edge Roman dominating function} of a graph $G$ is a function $f : E(G)\rightarrow \{0,1,2 \}$ such that every edge $e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e') = 2.$ The weight of an edge Roman dominating function $f$ is the value $w(f)=\sum_{e\in E(G)}f(e)$. The edge Roman domination number of $G$, denoted by $\gamma_R'(G)$, is the minimum weight of an edge Roman dominating function of $G$. In this paper, we characterize trees with edge Roman domination number twice the edge domination number.

For a graph G=V,E, a subset F of E is called an edge dominating set of G if every edge not in F is adjacent to some edge in F. The edge domination number γ′G of G is the minimum cardinality taken over all edge dominating sets of G. Here, we determine the edge domination number for shadow graphs, middle graphs, and total graphs of paths and cycles.

Pseudo-one-factorizations of regular graphs of odd order — II. products of graphs

This work presents the employment of graph theory for cryptographic applications. A Cartesian product graph (CPG) is used to propose new symmetric encryption schemes. On proposed encryption schemes, the ciphertexts of the plaintexts are sent to the receiver entity as the CPGs. The CPG is created based on two graphs G and H that are generated depending on the random choices of the plaintext bits. These bits are used as vertices of graphs G and H. The purpose of using the CPG is to increase the security level of the symmetric encryption schemes in comparison with other encryption schemes. Two study cases of the proposed symmetric Cartesian product graphic encryption (SCPGE) schemes are presented as new experimental results. The security issues of the proposed SCPGE schemes are determined. The SCPGE schemes consider as new insights for more secure communications.

This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least 2 k vertices is k -linked if, for every set of 2 k distinct vertices organised in arbitrary k pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We show that the Cartesian product K d 1 +1 × K d 2 +1 of complete graphs K d 1 +1 and K d 2 +1 is ⌊( d 1 + d 2 )/2⌋ -linked for d 1 , d 2 ≥ 2 , and this is best possible. This result is connected to graphs of simple polytopes. The Cartesian product K d 1 +1 × K d 2 +1 is the graph of the Cartesian product T ( d 1 ) × T ( d 2 ) of a d 1 -dimensional simplex T ( d 1 ) and a d 2 -dimensional simplex T ( d 2 ) . And the polytope T ( d 1 ) × T ( d 2 ) is a simple polytope , a ( d 1 + d 2 ) -dimensional polytope in which every vertex is incident to exactly d 1 + d 2 edges. While not every d -polytope is ⌊ d /2⌋ -linked, it may be conjectured that every simple d -polytope is. Our result implies the veracity of the revised conjecture for Cartesian products of two simplices.

The closed neighborhood $$N_G[e]$$NG[e] of an edge $$e$$e in a graph $$G$$G is the set consisting of $$e$$e and of all edges having a common end-vertex with $$e$$e. Let $$f$$f be a function on $$E(G)$$E(G), the edge set of $$G$$G, into the set $$\{-1, 1, 2\}$${-1,1,2}. If $$ \sum _{x\in N[e]}f(x) \ge 1$$?x?N[e]f(x)?1 for every edge $$e$$e of $$G$$G and every edge $$e$$e for which $$f (e) = -1$$f(e)=-1 is adjacent to at least one edge $$e'$$e? for which $$f (e')= 2$$f(e?)=2, then $$f$$f is called a signed Roman edge dominating function of $$G$$G. The minimum of the values $$\sum _{e\in E(G)} f(e)$$?e?E(G)f(e), taken over all signed Roman edge dominating functions $$f$$f of $$G$$G, is called the signed Roman edge domination number of $$G$$G and is denoted by $$\gamma _{sR}'(G)$$?sR?(G). In this note we initiate the study of the signed Roman edge domination in graphs and present some (sharp) bounds for this parameter.

A set is said to be a private edge dominating set, if it is an edge dominating set, for every has at least one external private neighbor in . Let and denote the minimum and maximum cardinalities, respectively, of a private edge dominating sets in a graph . In this paper we characterize connected graph for which ? q/2 and the graph for some upper bounds. The private edge domination numbers of several classes of graphs are determined.Keywords: Edge domination; Perfect domination; Private domination; Edge irredundant sets.© 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi: http://dx.doi.org/10.3329/jsr.v5i2.12024 J. Sci. Res. 5 (2), 283-294 (2013)

On the geodetic number and related metric sets in Cartesian product graphs

We present theoretical results when applying the Cartesian product of two Kuramoto models on different network topologies. By a detailed mathematical analysis, we prove that the dynamics on the Cartesian product graph can be described by the canonical equations as the Kuramoto model. We show that the order parameter of the Cartesian product is the product of the order parameters of the factors. On the product graph, we observe either continuous or discontinuous synchronization transitions. In addition, under certain conditions, the transition from an initially incoherent state to a coherent one is discontinuous, while the transition from a coherent state to an incoherent one is continuous, presenting a mixture state of first and second order synchronization transitions. Our numerical results are in a good agreement with the theoretical predictions. These results provide new insight for network design and synchronization control.

On the complexity of recognizing S-composite and S-prime graphs

In this paper we discuss the number of effective edges in the Cartesian product of fuzzy graphs. Some basic theorems and results are obtained on effective edges. Based on the results we find the effective edge domination number for Cartesian product of complete fuzzy graphs.

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes.Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.

Fiber-complemented graphs — I: structure and invariant subgraphs