Discrete Mathematics 219 (2000) 153–171 www.elsevier.com/locate/disc View metadata, citation and similar papers at core.ac.uk brought to you by CORE On optimal orientations of G vertex-multiplications provided by Elsevier - Publisher Connector K.M. Koh ∗, E.G. Tay Department of Mathematics, National University of Singapore, Lower Kent Ridge Road, Singapore 119260, Singapore Received 3 February 1999; revised 19 August 1999; accepted 7 September 1999 Abstract For a bridgeless connected graph G, let D(G) be the family of strong orientations of G; and for any D ∈ D(G), we denote by d(D) (resp., d(G)) the diameter of D (resp., G). Deÿne d(G)= min{d(D) | D ∈ D(G)}. In this paper, we study the problem of evaluating d(G(s1;s2;:::;sn)), where G(s1;s2;:::;sn)isaG vertex-multiplication for any connected graph G of order n¿3 and any sequence (si) with si¿2;i=1; 2;:::;n. While it is trivial that d(G)6d(G(s1;s2;:::;sn)), we show that d(G(s1;s2;:::;sn))6d(G) + 2. All graphs of the form G(s1;s2;:::;sn) are thus divided into the following three classes: Ci = {G(s1;s2;:::;sn) | d(G(s1;s2;:::;sn)) = d(G)+i}; i =0; 1; 2. We exhibit various families of graphs G(s1;s2;:::;sn)inCi for each i =0; 1; 2. c 2000 Elsevier Science B.V. All rights reserved. 1. Introduction Let G be a connected graph with vertex set V (G) and edge set E(G). For v ∈ V (G), the eccentricity e(v)ofv is deÿned as e(v) = max{d(v; x) | x ∈ V (G)}, where d(v; x) denotes the distance from v to x. The diameter of G, denoted by d(G), is deÿned as d(G) = max{e(v) | v ∈ V (G)}. Let D be a digraph with vertex set V (D) and edge set E(D). For v ∈ V (D), the notions e(v) and d(D) are similarly deÿned. An orientation of a graph G is a digraph obtained from G by assigning to each edge in G a direction. An orientation D of G is strong if every two vertices in D are mutually reachable in D. An edge e in a connected graph G is a bridge if G − e is disconnected. Robbins’ celebrated one-way street theorem [24] states that a connected graph G has a strong orientation if and only if no edge of G is a bridge. Ecient algorithms for ÿnding a strong orientation for a bridgeless connected graph can be found in Roberts [25], Boesch and Tindell [1] and Chung et al. [2]. Boesch and Tindell [1] extended Robbin’s result to mixed graphs where edges could be directed or undirected. Chung et al. [2] provided a linear-time algorithm for testing whether a ∗ Corresponding author. E-mail address: [email protected] (K.M. Koh). 0012-365X/00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved. PII: S0012-365X(99)00373-8 154 K.M. Koh, E.G. Tay / Discrete Mathematics 219 (2000) 153–171 mixed graph has a strong orientation and ÿnding one if it does. As another possible way of extending Robbins’ theorem, consider further the notion (G) given below (see [1,3,26]). Given a connected graph G containing no bridges, let D(G) be the family of strong orientations of G. Deÿne (G) = min{d(D) | D ∈ D(G)}−d(G): The ÿrst term on the right-hand side of the above equality is essential. Let us write d(G) = min{d(D) | D ∈ D(G)}: The problem of evaluating d(G) for an arbitrary connected graph G is very dicult. As a matter of fact, ChvÃatal and Thomassen [3] showed that the problem of deciding whether a graph admits an orientation of diameter two is NP-hard. On the other hand, the parameter d(G) has been studied in various classes of graphs including the cartesian product of graphs [22,31,20,27–30,8,11–18], complete graphs [21,1,19,23], complete bipartite graphs [22,1,31,5] and complete n-partitie graphs for n¿3 [22,5–7,9,10]. These optimal orientations can be used to provide optimal arrange- ments of one-way streets [24,27–30,12]. They also have applications for the gossip problem on a graph G, where all points simultaneously broadcast items to all other points in such a way that items are combined at no cost and all links are simultane- ously used but in only one direction at a time, because the time taken for the gossip to be completed is bounded above by min{2d(G); d(G)} (see [4]). In this paper, we shall extend the results on the complete n-partite graphs by intro- ducing a new family of graphs based on a given connected graph as follows. Let G be a given connected graph of order n with vertex set V (G)={v1;v2;:::;vn}. For any sequence of n positive integers (s ), let G(s ;s ;:::;s ) denote the graph with vertex i S 1 2 n ∗ ∗ ∗ n set V and edge set E such that V = i=1 Vi, where Vi’s are pairwise disjoint sets ∗ ∗ with |Vi| = si;i=1; 2;:::;n; and for any two distinct vertices x; y in V ;xy∈ E i x ∈ Vi and y ∈ Vj for some i; j; ∈{1; 2;:::;n} with i 6= j such that vivj ∈ E(G). We call the graph G(s1;s2;:::;sn)aG vertex-multiplication. Thus when G = Kn, the complete graph of order n, the graph G(s1;s2;:::;sn) is a complete n-partite graph. We ∼ call G a parent graph of a graph H if H = G(s1;s2;:::;sn) for some sequence (si)of positive integers. For convenience, we shall write, for i =1; 2;:::;n; Vi = {(p; i) | 16p6si} and ∗ call (p; i) the pth vertex in Vi. Thus two vertices (p; i) and (q; j)inV are adja- cent in G(s1;s2;:::;sn)i i 6= j and vivj ∈ E(G). For s =1; 2;:::; we shall denote G(s;s;:::;s) simply by G(s). Thus G(1) = G, and it is understood that the number of s0sinG(s;s;:::;s) is equal to the order of G. In this paper, we shall begin with the study on the relation between d(G(s1;s2;:::;sn)) and d(G). While it is trivial that d(G)6d(G(s1;s2;:::;sn)), we shall show that if si¿2 for each i =1; 2;:::;n where n¿3, then d(G(s1;s2;:::;sn))6d(G) + 2. This fundamen- tal result naturally gives rise to a 3-classiÿcation of graphs of the form G(s1;s2;:::;sn) according to d(G(s1;s2;:::;sn)) = d(G)+i; i =0; 1; 2. In the subsequent sections, we shall exhibit various families of graphs for each of the classes in the classiÿcation. K.M. Koh, E.G. Tay / Discrete Mathematics 219 (2000) 153–171 155 2. Terminology and some known results Let G be a connected graph of order n and F ∈ D(G(s1;s2;:::;sn)). For convenience, we write V (G)={1; 2;:::;n}. To avoid ambiguity, the vertex i is sometimes written as (i). We shall denote by NG(v) the set of vertices adjacent to v in G. Let A be a subdigraph of F. The eccentricity, outdegree and indegree of a vertex (p; i)inA are − denoted respectively by eA((p; i));sA((p; i)) and sA ((p; i)). The subscript A is omitted if A = F. Let D be a digraph. A dipath (resp., dicycle) in D is simply called a path (resp., cycle) in D. For X ⊆ V (D), the subdigraph of D induced by X is denoted by D[X ], or simply [X ], if there is no danger of confusion. Given F ∈ D(G(s ;s ;:::;s )), let S 1 2 n ik j=i Fj = F[{(p; j) | 16p6sj;j= i1;i2;:::;ik }]. 1 ∼ A digraph D1 is said to be isomorphic to a digraph D2, written D1 = D2, if there is a bijection ’ : V (D1) → V (D2) such that uv ∈ E(D1) if and only if ’(u)’(v) ∈ E(D2). For x; y; ∈ V (D), we write ‘x → y’or‘y ← x’ifx is adjacent to y in D. Also, for A; B ⊆ V (D), we write ‘A → B’or‘B ← A’ifx → y in D for all x ∈ A and for all y ∈ B. When A = {x}, we shall write ‘x → B’or‘B ← x’ for A → B. For x ∈ V (D), deÿne O(x)={y ∈ V (D) | x → y}. The following results on complete bipartite graphs K(p; q) and complete n-partite graph Kn(s1;s2;:::;sn) will be found useful in our discussion later. Theorem A (SoltÃÄ es [31] and Gutin [5]). For q¿p¿2, p 3 if q6 ; b p c (i) d(K(p; q)) = 2 p 4 if q¿ p ; b 2 c p 1 if q6 ; b p c (ii) (K(p; q)) = 2 p 2 if q¿ p ; b 2 c where bxc denotes the greatest integer not exceeding the real x. Theorem B (PlesnÃk [22] and Gutin [5]). For n¿3, (i) 26d(K(s1;s2;:::;sn))63; (ii) 06(K(s1;s2;:::;sn))61. 3. A fundamental result and classiÿcation of G(s1;s2;:::;sn) Throughout the remaining sections, let G be a connected graph of order n.