Many graphs such as hypercubes, star graphs, pancake graphs, grids, tori etc are known to be good interconnection network topologies. In any network topology, the vertices represent the processors and the edges represent links between the processors. Two most important criteriaefficiency and reliability of network models-can be studied with the help of graph theoretical techniques. The lexicographic product is a well studied graph product. The distance notions such as various diameters of a graph help to analyze the efficiency of any interconnection network. In this paper, we study some distance notions such as wide diameter, diameter variability and diameter vulnerability of lexicographic products that could be used in the design of interconnection networks.
The P 3 intersection graph of a graph G has for vertices all the induced paths of order 3 in G. Two vertices in P 3 (G) are adjacent if the corresponding paths in G are not disjoint. A w-container between two different vertices u and v in a graph G is a set of w internally vertex disjoint paths between u and v. The length of a container is the length of the longest path in it. The w-wide diameter of G is the minimum number l such that there is a w-container of length at most l between any pair of different vertices u and v in G. Interconnection networks are usually modeled by graphs. The w-wide diameter provides a measure of the maximum communication delay between any two nodes when up to w − 1 nodes fail. Therefore, the wide diameter constitutes a measure of network fault tolerance. In this paper we construct containers in P 3 (G) and apply the results obtained to the study of their connectivity and wide diameters.
The edge C 4 graph of a graph G, E 4 (G) is a graph whose vertices are the edges of G and two vertices in E 4 (G) are adjacent if the corresponding edges in G are either incident or are opposite edges of some C 4. In this paper, we show that there exist infinitely many pairs of non isomorphic graphs whose edge C 4 graphs are isomorphic. We study the relationship between the diameter, radius and domination number of G and those of E 4 (G). It is shown that for any graph G without isolated vertices, there exists a super graph H such that C(H) = G and C(E 4 (H)) = E 4 (G). Also we give forbidden subgraph characterizations for E 4 (G) being a threshold graph, block graph, geodetic graph and weakly geodetic graph.
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