The Computational Complexity of Deciding

We determine the computational complexity of several dmision p r o b lerns related to hamiltonian-connected graphs. In particular, we show that it is NP-complete to determine whether a graph G is harniltonian-connected. We also show that it is NP-complete to determine whether G is hamiltonian-connerted from a distinguished vertex u . Lastly, we consider the complexity of multiple-solution and unique-solution variations. The notion of a hamilfonian-connected graph, in which there is a hamiltonian path between each pair of distinct vertices, was introduced in 1963 by Ore (111. In 1981 Chartrand and Nordhaus [2] defined a variant of this idea: a graph G is hamillonian-connected from a ve r f e z u if there are hamiltonian paths from v to each other vertex of G. Hendry [5] studied graphs uniquely hamiltonian-connected from a vertex u , in which there is a unique hamiltonian path from v to each other vertex of G. While considerable work has been done on the structure of such graphs 12, 6, 10, 31, very little has been said about the computational complexity of the associated decision problems, despite the fact that the decision problems for hamiltonian and related graphs are among the most famous NP-complete problems. In this paper we examine the complexity of several of these problems. We show in Section 2 that it is NP-complete to determine whether a graph is hamiltonianconnected. In Section 3 we show that it is NP-complete to determine if a graph is hamiltonian-connected from a vertex. In Section 4 we examine the complexity of multiply hamiltonian-connected problems, and, in Section 5, unique hamiltonianconnected problems. In this section we will refer to the following decision problems. CONGRESSUS NUMERANTIUM 93(1993), pp.209-214 HCON Instance: A graph G. Question: Is G hamiltonian-connected? Huv Instance: A graph G and distinct vertices u, v. Question: Is there a hamiltonian path from u to U P Huvl Instance: A graph G and distinct, degree 1 vertices u, v. Question: Is there a hamiltonian path from u to vP These three problems are all obvious members of NP. The hamiltonian path problem, Huv, is NP-complete [4] . The NP-completeness of Huvl follows easily, and is shown below. We then give a polynomial transformation from Huvl to HCON, showing that HCON is also NP-complete. Lemma 1. Huvl is NP-complete. Proof. We give a transformation from Huv. Let (G, u , v) be an instance of HUV. Construct a new graph G' by appending degree 1 verticeu, u' to u and v' to v. It is ciear that ( G , u , v) is a yes-inst,mce of Huv if and only if (G', u', v') is a yes-instance ofG ' . Theorem 1. HGON is NP-complete. Proof. We give a transformation from Huvl . Let (G. u , v) be an instance of Huvl . so that u and v each have degree 1. Construct a new graph G I by adding two new vertices, vl and v2, with vl adjacent t o u , v, and 02, and v2 adjacent t o all other vertices in G I . We claim that G has a hamiltonian u-v path if and only if G I is hamiltonian-connected. First, let H be a hamiltonian u-v path in G. For a vertex u1 in G , we will denote by w, (resp. w,) the neighbor of w on H that is closer t o u (resp. v). We will also denote by H ( w , z ) the section of H from the vertex w t o the vertex z . We show below how to use H to construct hamiltonian paths between each pair of vertices in G I . We use z and y to denote vertices of G I , not equal to rr or v, with z closer than y to u on the hamiltonian path H.