Some Results On Convex Greedy Embedding Conjecture for 3-Connected Planar Graphs

A greedy embedding of a graph $G = (V,E)$ into a metric space $(X,d)$ is a function $x : V(G) \to X$ such that in the embedding for every pair of non-adjacent vertices $x(s), x(t)$ there exists another vertex $x(u)$ adjacent to $x(s)$ which is closer to $x(t)$ than $x(s)$. This notion of greedy embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci. 2005), where authors conjectured that every 3-connected planar graph has a greedy embedding (possibly planar and convex) in the Euclidean plane. Recently, greedy embedding conjecture has been proved by Leighton and Moitra (FOCS 2008). However, their algorithm do not result in a drawing that is planar and convex for all 3-connected planar graph in the Euclidean plane. In this work we consider the planar convex greedy embedding conjecture and make some progress. We derive a new characterization of planar convex greedy embedding that given a 3-connected planar graph $G = (V,E)$, an embedding x: V \to \bbbr^2 of $G$ is a planar convex greedy embedding if and only if, in the embedding $x$, weight of the maximum weight spanning tree ($T$) and weight of the minimum weight spanning tree (\func{MST}) satisfies \WT(T)/\WT(\func{MST}) \leq (\card{V}-1)^{1 - \delta}, for some $0 < \delta \leq 1$.Comment: 19 pages, A short version of this paper has been accepted for presentation in FCT 2009 - 17th International Symposium on Fundamentals of Computation Theor

On Claw- And Net-Free Graphs

R u t c o

On the distribution of eigenvalues of graphs

Let G be a simple undirected graph with n greater than or equal to 2 vertices and let alpha(0)(G) greater than or equal to ..., alpha(n-1)(G) be the eigenvalues of the adjacency matrix of G, It is shown by Cao and Yuen (1995) that if alpha(1)(G) = - 1 then G is a complete graph, and therefore alpha(0)(G) = n - 1 and alpha(i)(G) = -1 for 1 less than or equal to i less than or equal to n - 1. We obtain similar results for graphs whose complement is bipartite. We show in particular, that if the complement of G is bipartite and there exists an integer k such that 1 less than or equal to k < (n - 1)/2 and alpha(k)(G) = -1 then alpha(i)(G) = -1 for k less than or equal to i less than or equal to n - k + 1. We also compare and discuss the relation between some properties of the Laplacian and the adjacency spectra of graphs. (C) 1999 Elsevier Science B.V. All rights reserved