Finding the longest isometric cycle in a graph

When searching for the longest cycle in a graph, we sometimes want to restrict our search space to the cycles that do not have crossing edges induced cycles. If we think of the crossing edges as shortcuts, looking for induced cycles is then looking for shortcut-free cycles. However, an induced cycle may have a shortcut in the following sense: For two vertices on the cycle, the distance between them in the graph is strictly less than the distance between them in the cycle. As an example, consider the wheel graph on 7 or more vertices. The circumference of the wheel induces a cycle. In this cycle, the distance between diametrically opposite vertices is at least 3. At the same time one can get between any two vertices on this cycle in two steps via the central node. Thus this cycle is induced, but clearly not shortcut-free. We will say that a cycle that has no shortcuts in the above sense is an isometric cycle of G. Finding the longest isometric cycle of a graph is then a natural variant of the problem of finding a longest cycle. In this paper we present a polynomial time algorithm for finding the longest isometric cycle in a graph. 1 Definitions and terminology All input graphs are simple, connected, unweighted and undirected. A walk W is a sequence of vertices where each consecutive pair of vertices is connected by an edge. If the first and last vertex of W are the same we say that W is cyclic. If all vertices in W are unique we say that the walk is a path. If W has at least 3 vertices and all vertices of W are unique, but the first and last vertex are the same, W is a cycle. The length of a walk is the number of edges in it. The number of edges in the shortest path between two vertices u and v in a graph G is denoted dG(u, v) and is called the distance between u and v. When the graph is not specified we implicitly mean distance in G and will write d(u, v) for short. If u and v actually are the same vertex, we say that dG(u, v) = 0. If u and v lie in different components of G, the distance between them is infinite. For a natural number p, G to the power of p is the graph G = (V (G), {(u, v) : dG(u, v) ≤ p}). A subgraph H of a graph G is an isometric subgraph, if for every u and v in V (H), dH(u, v) = dG(u, v). Notice that an isometric subgraph is an induced subgraph. 2 A useful auxiliary graph In this section we are going to concentrate upon an auxiliary graph that we can use to test whether a given graph G has an isometric cycle of length exactly k. We will assume that k ≥ 3. Using G we build an new graph Gk. The set of vertices of Gk is the set of ordered pairs {(u, v) ∈ V : d(u, v) = bk/2c} and there is an edge between (u, v) and (w, x) in Gk if (u,w) ∈ E(G) and (v, x) ∈ E(G). In order to use Gk we ∗Department of Informatics, University of Bergen, N-5020 Bergen, Norway. Email: [email protected].