tailieunhanh - Báo cáo toán học: "Induced paths in twin-free graphs"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Induced paths in twin-free graphs. | Induced paths in twin-free graphs David Auger Telecom ParisTech 46 rue Barrault 75634 Paris Cedex 13 France auger@ Submitted Feb 19 2008 Accepted May 27 2008 Published Jun 6 2008 Mathematics Subject Classification 05C12 Abstract Let G V E be a simple undirected graph. Given an integer r 1 we say that G is r-twin-free or r-identifiable if the balls B v r for v 2 V are all different where B v r denotes the set of all vertices which can be linked to v by a path with at most r edges. These graphs are precisely the ones which admit r-identifying codes. We show that if a graph G is r-twin-free then it contains a path on 2r 1 vertices as an induced sugbraph . a chordless path. keywords graph theory identifying codes twin-free graphs induced path radius 1 Notation and definitions Let G V E be a simple undirected graph. We will denote an edge x y 2 E simply by xy. A path in G is a sequence P v0V1 Vk of vertices such that for all 0 i k 1 we have vivi 1 2 E if Vo x and Vk y we say that P is a path between x and y. The length of a path P v0V1 vk is the number of edges between consecutive vertices . k. If x y 2 V we define the distance d x y to be the minimum length of a path between x and y. Then a shortest path between x and y is a path between x and y of length precisely d x y . If r 0 B x r will denote the ball of centre x and radius r which is the set of all vertices V of G such that d x v r. If P v0 vk is a path in G a chord in P is any edge viVj 2 E with i j 1 1. A path is chordless if it has no chord in this case there is an edge between two vertices of the path vi and Vj if and only if i and j are consecutive . i j 1. It is straightforward to see that any shortest path is chordless. If x 2 V we define the eccentricity of x by ecc x max d x v . v2V THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N17 1 The diameter of G is the maximum eccentricity of a vertex in G whereas the radius rad G of G is the minimum eccentricity of a vertex in G. A vertex x such