A survey on pairwise compatibility graphs

Abstract

Let T be an edge weighted tree and d min , d max be two non-negative real numbers where d min d max : The pairwise compatibility graph (PCG) of T for d min , d max is a graph G such that each vertex of G corresponds to a distinct leaf of T and two vertices are adjacent in G if and only if the weighted distance between their corresponding leaves lies within the interval ½d min , d max : A graph G is a PCG if there exist an edge weighted tree T and suitable d min , d max such that G is a PCG of T. The class of pairwise compatibility graphs was introduced to model evolutionary relationships among a set of species. Since not all graphs are PCGs, researchers become interested in recognizing and characterizing PCGs. In this paper, we review the results regarding PCGs and some of its variants.

An Efficient Shortest Path Algorithm: Multi-Destinations in an Indoor Environment

“…Figure 20 shows the visited-wise comparison among EAMDSP, CDSSSD and MDMSMD as per Table 22. Finally, based on Figures 13,14,19,and 20,and Tables 13,14,21,and 22 it is reported that the proposed EAMDSP demonstrates the outstanding performance on cost and number of visited nodes compared to CDSSSD and MDMSMD.…”

“…As of now, Dijkstra's is a renowned path searching algorithm that is mainly used for searching the shortest path from a single source to a single destination introduced by Edsger W. Dijkstra in 1956. A typical single-source single-destination, Dijkstra's algorithm, is used for single-pair shortest path problems [14]. It keeps up a set S of unraveled nodes, consisting of those hubs or nodes whose ultimate shortest path between the starting node verses has ascertained as well as labels t(i), conserving the upward bound of the smallest pathway distance among v s and v i [15].…”

All Graphs with at most 8 nodes are 2-interval-PCGs

“…Since then, much work has been done in order to understand which graphs are PCGs (e.g. see the surveys [3,4]).…”

New Results on Pairwise Compatibility Graphs

“…Constructing a PCT from a given graph G for a suitable d min and d max is an interesting problem and a lot of research works have been conducted on identifying different classes of graphs as PCGs like complete graphs, trees, cycles, block-cycle graphs, interval graphs, threshold graphs, threshold tolerance graphs, split matching graphs, ladder graphs, outer subdivision of lad-der graphs, triangle-free outer planar 3-graphs, Dilworth-1 graphs, Dilworth-2 graphs, every graph of at most seven vertices etc [3,6,7,19,20,21,22,23,24]. Subsequently, some other graphs are also discovered which are not PCGs like wheel graphs with at least 9 vertices, certain bipartite graphs etc [25,26,27].…”

“…It is still unknown whether grid graphs are PCGs or not. But it has been proven that disk graphs, a superclass of grid graphs, are not PCGs [23] and ladder graphs, a subclass of grid graphs, are PCGs [19]. The PCT of a ladder graph, as constrcuted by [19], is a centepede where the order of leaves connected to the spine is the row major order of the original ladder graph.…”