Average distance is an important parameter for measuring the communication cost of computer networks. A popular approach for its computation is to first partition the edge set of a network into convex components using the transitive closure of the Djoković-Winkler’s relation and then to compute the average distance from the respective invariants of the components. In this paper we refine this idea further by shrinking the quotient graphs into smaller weighted graph called reduced graph, so that the average distance of the original graph is obtained from the reduced graphs. We demonstrate the significance of this technique by computing the average distance of butterfly and hypertree architectures. Along the way a computational error from [European J. Combin. 36 (2014) 71–76] is corrected.

Average Distance in Interconnection Networks via Reduction Theorems for Vertex-Weighted Graphs

Locating defective elements in a processor network brings the motivation for location dominating sets. In this paper, the location dominating set problem for friendship graphs is solved. Location problems examine the ability to pinpoint the origin of an event. Friendship graphs are used to analyze social networks. Domination in social networks is used to find the most powerful node in such networks and location domination is significant in locating the faulty or the unusable node in that network.

Location Domination for Generalized Friendship Graphs Analytics

Let $$G=(V,E)$$ be a graph with no isolated vertex. A subset $$S\subseteq V(G)$$ is a total dominating set of graph G if every vertex in V(G) is adjacent to at least one vertex in S. A total dominating set S of graph G is a locating-total dominating set if for every pair of distinct vertices $$u_1$$ and $$u_2$$ in $$V(G)-S$$, $$N(u_1)\cap S\ne N(u_2)\cap S$$. The locating-total domination number of graph G, denoted by $$\gamma _t^L(G)$$, is the minimum cardinality of a locating-total dominating set of G. In this paper, we investigate the bounds of locating-total domination number of grid graphs.

Locating-Total Domination in Grid Graphs

Domination Parameters in Hypertrees and Sibling trees

A note on the complexity of locating-total domination in graphs

Abstract In this paper we obtain a sharp (improved) lower bound on the locating-total domination number of a graph, and show that the decision problem for the locating-total domination is NP-complete.

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A Note on the Locating-Total Domination in Graphs

Combinatorial Properties of Root-fault Hypertrees☆

A locating-dominating set (LDS) of a graph $G$ is a dominating set $S$ of $G$ such that for every two vertices $u$ and $v$ in $V(G) \setminus S$, $N(u)\cap S \neq N(v)\cap S$. The locating-domination number $\gamma^{L}(G)$ is the minimum cardinality of a LDS of $G$. Further if $S$ is a total dominating set then $S$ is called a locating-total dominating set. In this paper we determine the domination, total domination, locating-domination and locating-total domination numbers for hypertrees and sibling trees.

A locating-dominating set LDS S of a graph G is a dominating set S of G such that for every two vertices u and v in $$VG \setminus S$$, $$Nu\cap S \ne Nv\cap S$$. The locating-domination number $$\gamma ^{L}G$$ is the minimum cardinality of a LDS of G. Further if S is a total dominating set then S is called a locating-total dominating set. In this paper we determine the domination, total domination, locating-domination and locating-total domination numbers for hypertrees.

R. Jayagopal

Papers

A Note on the Locating-Total Domination in Graphs

Combinatorial Properties of Root-fault Hypertrees☆

Domination Parameters in Hypertrees and Sibling trees

Domination Parameters in Hypertrees and Sibling trees

Domination Parameters in Hypertrees