Induced Paths in Twin-Free Graphs

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 ∈ 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, i.e. 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} ∈ 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 ∈ E; if v0 = 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, i.e. k. If x, y ∈ 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 ∈ E with |i − j| 6= 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.e. |i − j| = 1. It is straightforward to see that any shortest path is chordless. If x ∈ V , we define the eccentricity of x by ecc(x) = max v∈V d(x, v). 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 that ecc(x) = rad(G) is a centre of G. So G has radius t ≥ 1 and x is a centre of G if and only if B(x, t) = V whereas B(v, t − 1) 6= V for all v ∈ V . If W ⊂ V , the sugbraph of G induced by W is the graph whose set of vertices is W and whose edges are all the edges xy ∈ E such that x and y are in W . We denote this graph by G[W ]; if W = V \ {v}, we simply write G[V − v]. An induced path in G is a subset P of V such that G[P ] is a path; equivalently, the vertices in P define a chordless path in G. All these terminology and notation being standard, we refer to [3] for further explanation. Two distinct vertices x and y are called r-twins if B(x, r) = B(y, r). If there are no r-twins in G, we say that G is r-twin-free. 2 Motivations and main results The notion of identifying code in a graph was introduced by Karpovsky, Chakrabarty and Levitin in [5]. For r ≥ 1, an r-identifying code in G = (V, E) is a subset C of V such that the sets IC(v) = B(v, r) ∩ C for v ∈ V are all distinct and non-empty. The original motivation for identifying codes was the fault diagnosis in multiprocessor systems; we refer to [1], [5] or [7] for further explanation and applications. The interested reader can also find a nearly exhaustive bibliography in [6]. Given a graph G = (V, E), it is easily seen that there exists an r-identifying code in G if and only if V itself is an r-identifying code, which precisely means that G is r-twinfree. Different structural properties which are worth investigating arise when considering a connected r-twin-free graph with r ≥ 1. For instance, it has been proved in [2] that an r-twin-free graph always contains a path, not necessarily induced, on 2r + 1 vertices. In the same article, the authors conjectured that we can always find such a path as an induced subgraph of G. We prove this conjecture as a corollary from Theorem 1. Let us denote by p(G) the maximum number of vertices of an induced path in G. We prove the following theorem and corollary, which we formulate for connected graphs without loss of generality. Theorem 1. Let G = (V, E) be a connected graph with at least two vertices, and with a centre c ∈ V such that no neighbour of c is a centre. Then p(G) ≥ 2 rad(G) + 1. This implies: Corollary 2. Let G be a connected graph with at least two vertices, and r ≥ 1. If G is r-twin-free then p(G) ≥ 2r + 1. the electronic journal of combinatorics 15 (2008), #N17 2 3 Proof of the theorem A different proof for Corollary 2 can be found in [1]. The one we present here is much shorter and is based on the article by Erdős, Saks and Sós [4] where the following theorem can be found. The authors give credit to Fan Chung for the proof. Theorem 3. (Chung) For every connected graph G = (V, E) we have p(G) ≥ 2 rad(G) − 1. We require the following lemma, inspired by [4], in order to prove Theorem 1. Lemma 4. Let t ≥ 2 and G = (V, E) be a graph such that there are in G two vertices v0 and vt with d(v0, vt) = t, a shortest path v0v1v2 · · ·vt between v0 and vt, and a vertex w such that d(v0, w) ≤ t− 1 and d(v2, w) ≥ t (see fig. 1). Then there exists an induced path on 2t − 1 vertices in G. v0 v1 v2 vt w d(v0,w)≤t−1 d(v2 ,w)≥t Figure 1: The path v0 · · · vt and w in Lemma 4. Proof. In the case t = 2, the shortest path v0v1v2 itself is an induced path on 2t − 1 = 3 vertices; so we suppose now that t ≥ 3. First observe that since d(v2, w) ≥ t we have w 6= vi for all i ∈ {0, 1, · · · , t}. Consider a shortest path P between v0 and w, and let u ∈ P , distinct from v0. Let i ≥ 2; we show that d(u, vi) ≥ 2. First we have d(v0, vi) = i ≤ d(v0, u) + d(u, vi) and second t ≤ d(v2, w) ≤ d(v2, vi) + d(vi, u) + d(u, w) with d(v2, vi) = i − 2 because i ≥ 2. Summing these two inequalities we get t + i ≤ d(v0, u) + d(u, w) + 2d(vi, u) + i − 2 and since d(v0, u) + d(u, w) = d(v0, w) we deduce t + 2 ≤ d(v0, w) + 2d(u, vi). the electronic journal of combinatorics 15 (2008), #N17 3 But we have d(v0, w) ≤ t − 1 and so d(u, vi) ≥ 3 2 . Let us note that since d(v2, w) ≥ t, we have d(v0, w) ≥ t − 2 and so P consists of v0 and at least t− 2 ≥ 1 other vertices, i.e. at least t− 1 vertices. We proved that u satisfies d(u, vi) ≥ 2 for i ≥ 2, so u is distinct from all the vi’s and furthermore can be adjacent only to v1 or v0 (see fig. 2). v0 v1 v2 vt