Given a graph G, the (graph theory) general position problem is to find the maximum number of vertices such that no three vertices lie on a common geodesic. This graph invariant is called the general position number (gp-number for short) of G and denoted by gp( G). In this paper, the gp-number is determined for a large class of subgraphs of the infinite grid graph and for the infinite diagonal grid. To derive these results, we introduce monotone-geodesic labeling and prove a Monotone Geodesic Lemma that is in turn developed using the Erdös-Szekeres theorem on monotone sequences. The gp-number of the 3-dim infinite grid is bounded. Using isometric path covers, the gp-number is also determined for Beneš networks.

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 article, 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 Klavžar and Nadjafi-Arani ((2014) Wiener index in weighted graphs via unification of Θ*-classes⁠, Eur. J. Combin. 36, 71–76) is corrected.

Wiener index of a graph | $G$ | is defined as | $W(G) = \frac {1}{2} \sum _{{u,v \in V(G)}} d_{{G}}(u,v)$ | . The Transmission index | $T(u)$ | of a vertex | $u$ | in a graph | $G$ | is defined as | $T(u) = \sum _{{v \in V}}d(u,v)$ | . The original technique for the computation of Wiener index was by brute-force method applying distance matrix. Later a new technique using convex partition was introduced and this convex partition method was shown to be more efficient than distance matrix method. However, this convex partition method is not universal. Some interesting architectures such as butterfly and mesh of trees do not induce convex partition. In this paper, we introduce another partition technique to accommodate larger classes of graphs which are not solved by convex partition method. This partition technique is called transmission partition method. It is different from distance matrix method and convex partition method. We show that this new technique significantly reduces the time complexity to compute the Wiener index to constant time for larger classes of graphs. We demonstrate the efficiency of this technique on butterfly networks by computing its Wiener index and its Transmission index in constant time.

Graph embedding problems have gained importance in the field of interconnection networks for parallel computer architectures. Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. In this paper, we introduce a technique to obtain a lower bound for dilation of an embedding. Moreover, we give algorithms to compute exact dilation of embedding circulant network into a triangular grid, Tower of Hanoi graph and Sierpinski gasket graph, proving that the lower bound obtained is sharp.

Abstract We estimate and characterize the edge congestion‐sum measure for embeddings of various graphs such as cycles, wheels, and generalized wheels into arbitrary trees. All embedding algorithms apply an interesting general technique based on the consecutive label property. Our algorithms produce optimal values of sum of dilations and sum of edge‐congestions in linear time. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(3), 173–178 2004