If r 1, and m and n are each a multiple of (r + 1)2 + r2, then each isomorphic component of Cm ×Cn admits of a vertex partition into (r + 1)2 + r2 perfect r-dominating sets. The result induces a dense packing of Cm × Cn by means of vertexdisjoint subgraphs, each isomorphic to a diagonal array. Areas of applications include efficient resource placement in a diagonal mesh and error-correcting cod...

For a given number of colours, s, the guessing number of a graph is the base s logarithm of the size of the largest family of colourings of the vertex set of the graph such that the colour of each vertex can be determined from the colours of the vertices in its neighbourhood. An upper bound for the guessing number of the n-vertex cycle graph Cn is n/2. It is known that the guessing number equal...

We present a general framework for solving parameterized problems on weighted graphs. We use this framework to obtain efficient algorithms for such fundamental problems as Vertex Cover, 3-Hitting Set, Edge Dominating Set and k-Internal Out-Branching, on weighted graphs. For each of these problems, given an instance of size n and a weight parameter W ≥ 1, we seek a solution of weight at most (or...

The topology of an interconnection network can be modeled by a graph G=(V(G),E(G)). connectivity G is parameter used to measure the reliability corresponding network. direct product important product. This paper mainly focuses on super connectedness graphs and cycles. G, denoted κ(G), size minimum vertex set S⊆V(G) such that G−S not connected or has only one vertex. said connected, simply super...

Let G = (V, E) be a graph. A subset D of V is called common neighbourhood dominating set (CN-dominating set) if for every v ∈ V −D there exists a vertex u ∈ D such that uv ∈ E(G) and |Γ(u, v)| > 1, where |Γ(u, v)| is the number of common neighbourhood between the vertices u and v. The minimum cardinality of such CN-dominating set denoted by γcn(G) and is called common neighbourhood domination n...

We investigate hamiltonian properties of Pm × Cn, m ≥ 2 and even n ≥ 4, which is bipartite, in the presence of faulty vertices and/or edges. We show that Pm×Cn with n even is strongly hamiltonianlaceable if the number of faulty elements is one or less. When the number of faulty elements is two, it has a fault-free cycle of length at least mn−2 unless both faulty elements are contained in the sa...

A tournament T is an orientation of a complete graph, and a feedback vertex set of T is a subset of vertices intersecting every directed cycle of T . We prove that every tournament on n vertices has at most 1.6740 minimal feedback vertex sets and some tournaments have 1.5448 minimal feedback vertex sets. This improves a result by Moon (1971) who showed upper and lower bounds of 1.7170 and 1.475...

The Ramsey number R(Cn, Km) is the smallest integer p such that any graph G on p vertices either contains a cycle Cn with length n or contains an independent set with order m. In this paper we prove that R(Cn , K 4 ) = 3(n 1) + 1 (n ~ 4). We shall only consider graphs without multiple edges or loops. The Ramsey number R(Cn,Km) is the smallest integer p such that any graph G on p vertices either...

There are several known exact results on the crossing numbers of Cartesian products of paths, cycles or stars with “small” graphs. Let H be the 5-vertex graph defined from K5 by removing three edges incident with a common vertex. In this paper, we extend the earlier results to the Cartesian products of H ×Pn and H ×Cn, showing that in the general case the corresponding crossing numbers are 3n−1...