Paths, Cycles, and Connectivity

Abstract

In this chapter, we study some important fundamental concepts of graph theory. In Section 3.1 we start with the definitions of walks, trails, paths, and cycles. The well-known Eulerian graphs and Hamiltonian graphs are studied in Sections 3.2 and 3.3, respectively. In Section 3.4, we study the concepts of connectivity and connectivity-driven graph decompositions.

Exercises

• 1. Let G be a graph having exactly two vertices u and v of degree three and all other vertices have even degree. Then show that there is an u, v-path in G.

• 2. Write an algorithm to find a Eulerian circuit in an Eulerian graph based on the sufficiency proof of Lemma 3.2.1.

• 3. Give a proof of Lemma 3.2.2.

• 4. Count the minimum number of edges required to add for making a non-Eulerian graph an Eulerian graph.

• 5. Let $K_n$ be a complete graph of n vertices where n is odd and $n\ge 3$. Show that $K_n$ has $(n-1)/2$ edge-disjoint Hamiltonian cycles.

• 6. Show that every k-regular graph on $2k+1$ vertices is Hamiltonian [7].

• 7. Write an algorithm to find a pair of vertex disjoint paths between a pair of vertices in a 2-connected graph.

• 8. Write an algorithm to find an ear decomposition of a biconnected graph.

• 9. Let s be a designated vertex in a connected graph $G=(V,E)$. Design an $O(n\,{+}\,m)$ time algorithm to find a path between s and v with the minimum number of edges for all vertices $v\in V$.

• 10. Show that a 3-connected graph has at least six edges.