Minimum diameter and cycle-diameter orientations on planar graphs

Let G be an edge weighted undirected graph. For every pair of nodes consider the shortest cycle containing these nodes in G. The cycle diameter of G is the maximum length of a cycle in this set. Let H be a directed graph obtained by directing the edges of G. The cycle diameter of H is similarly defined except for that cycles are replaced by directed closed walks. Is there always an orientation H of G whose cycle diameter is bounded by a constant times the cycle diameter of G? We prove this property for planar graphs. These results have implications on the problem of approximating an orientation with minimum diameter. 1 Hereditary order on cycles Let G = (V,E) be a 2-edge connected undirected graph. Choose one of its nodes, mark it as z. For every node v ∈ V \{z} find the shortest undirected cycle connecting v and z, mark this cycle as C(v), we say that v is served by C(v). Let C = {C(v)|v ∈ V } and let GC be the graph induced by the edges in C. Let V1, . . . , Vl be the node sets of the connected components of GC induced by V \{z}. Let Gi i = 1, . . . , l be the subgraphs of GC induced by Vi ∪ {z}. Each of theses subgraphs will be oriented independently of the others. Clearly, the bound holds for the whole graph if it holds for every component. Hence w.l.o.g we can assume that l = 1. In Figure 1 G1, G2 and G3 are illustrated. For every edge e ∈ E let l(e) ≥ 0 be the length of the edge. 1. A path P is an ordered set of nodes (v1, . . . , vn) and distinct edges (v1, v2), (v2, v3), . . . , (vn−1, vn). When v1 = vn, P is a cycle. 2. For every G a subgraph of G, let E(G) (V (G)) be the subgraph’s edge set (node set). 3. For every path P , l(P ) = ∑ e∈E(P ) l(e). This is part of the Ph.D. dissertation “Approximation algorithms for three optimization problems on graphs” by Nili Guttmann-Beck, 2005. The special case of series-parallel graphs is solved in [1]. Department of Computer Science, The Academic College of Tel-Aviv Yaffo, Yaffo, Israel. Email: [email protected] ; School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel. Email: [email protected].