Algorithms for enumerating and counting D2CS of some graphs

TL;DR

This paper introduces algorithms for counting and enumerating D2CS (diameter 2 connected subgraphs) in various graphs, providing explicit formulas and efficient algorithms, including a linear time method for strongly chordal graphs.

Contribution

It presents new formulas for D2CS counts in specific trees and develops algorithms for enumeration and counting, notably a linear time algorithm for strongly chordal graphs.

Findings

Explicit formulas for D2CS in specific trees

Efficient enumeration and counting algorithms

Linear time algorithm for maximal D2CS in strongly chordal graphs

Abstract

A D2CS of a graph G is a set $S \subseteq V(G)$ with $diam(G[S]) \leq 2$. We study the problem of counting and enumerating D2CS of a graph. First we give an explicit formula for the number of D2CS in a complete k-ary tree, Fibonacci tree, binary Fibonacci tree and the binomial tree. Next we give an algorithm for enumerating and counting D2CS of a graph. We then give a linear time algorithm for finding all maximal D2CS in a strongly chordal graph.