On Existence of Must-Include Paths and Cycles in Undirected Graphs

TL;DR

This paper investigates the existence of specific paths and cycles in undirected graphs that must include certain vertices or edges, providing conditions and algorithms for these problems.

Contribution

It introduces necessary and sufficient conditions for must-involve paths and cycles, along with efficient algorithms to determine their existence.

Findings

Characterization of conditions for must-involve paths and cycles

Development of algorithms for these problems

Efficient solutions for related graph problems

Abstract

Given an undirected graph $G=(V,E)$ and vertices $s,t,w_1,w_2\in V$, we study finding whether there exists a simple path $P$ from $s$ to $t$ such that $w_1,w_2 \in P$. As a sub-problem, we study the question: given an undirected graph and three of its edges, does there exist a simple cycle containing all those edges? We provide necessary and sufficient conditions for the existence of such paths and cycles, and develop efficient algorithms to solve this and related problems.