A twin dominating set of a digraph D is a subset S of vertices if, for every vertex u ∉ S, there are vertices x,y ∈ S such that ux and yu are arcs of D. A digraph D is round if the vertices can be labeled as v0,v1,...,vn-1 so that, for each vertex vi, the out-neighbors of vi appear consecutively following vi and the in-neighbors of vi appear consecutively preceding vi. In this paper, we give polynomial time algorithms for finding a minimum weight twin dominating set and a minimum weight total twin dominating set for a weighted round digraph. Then we show that there is a polynomial time algorithm for deciding whether a locally semicomplete digraph has an independent twin dominating set. The class of locally semicomplete digraphs contains round digraphs as a special case.

Multi-source broadcasting is one of the information dissemination problems on interconnection networks such that some (but not all) units disseminate distinct information to all other units. In this paper, we discuss multi-source broadcasting on the Kautz digraph which is one of the models of interconnection networks. We decompose the Kautz digraph K(d,n) into isomorphic cycle-rooted trees whose root-cycle has length 2, then we present an algorithm for multi-source broadcasting using these cycle-rooted trees. This algorithm is able to treat d(d+1) messages simultaneously and takes the same order for required times as lower bound.

Multi-source broadcasting is one of the information dissemination problems on communication networks such that some units disseminate distinct messages to all other units. In this paper, we study multi-source broadcasting on the de Bruijn and Kautz digraphs which are the models of interconnection networks. In [8] and [12], a cycle-rooted tree which has a large root-cycle is constructed by composition of isomorphic factors, and the multi-source broadcasting is executed on the cycle-rooted tree. On the other side, we execute multi-source broadcasting on each isomorphic factors at the same time. We present a method for multi-source broadcasting using isomorphic cycle-rooted trees which factorize these digraphs, and investigate its efficiency.

In this paper, we present a new extension of the butterfly digraph, which is known as one of the topologies used for interconnection networks. The butterfly digraph was previously generalized from binary to d-ary. We define a new digraph by adding a signed label to each vertex of the d-ary butterfly digraph. We call this digraph the dihedral butterfly digraph and study its properties. Furthermore, we show that this digraph can be represented as a Cayley graph. It is well known that a butterfly digraph can be represented as a Cayley graph on the wreath product of two cyclic groups [1]. We prove that a dihedral butterfly digraph can be represented as a Cayley graph in two ways.

Let f and g be two maps from a set E into a set F such that f(x) g(x) for every x in E. Sahili [8] has shown that, if min {|f-1(z)|,|g-1(z)|}≤ n for each z∈ F, then E can be partitioned into at most 2n+1 sets E1,..., E2n+1 such that f(Ei)∩ g(Ei)= for each i=1,..., 2n+1. He also asked if 2n+1 is the best possible bound. By using Sahili's formulation of the problem in terms of the chromatic number of line digraphs, we improve the upper bound from 2n+1 to O(log n). We also investigate extended version for more than two maps.

We show that the necessary and sufficient condition for the existence of a balanced bowtie decomposition of the symmetric complete multi-digraph is n 5 and λ(n-1) 0 (mod 6). Decomposition algorithms are also given.

We define a d-matched digraph and propose a recursive procedure for designing an optimal d-matched digraph without bidirectional edges. The digraph represents an optimal highly structured system which is a special class of self-diagnosable systems and identifies all of the faulty units independently and locally in O(|E|) time complexity. The procedure is straightforward and gives a system flexible in network connections. Hence the procedure is applicable to real systems such as the Internet or cooperative robotic systems which change their topology dynamically.

There are several kinds of coloring of digraphs, such as vertex-coloring and arc-coloring. We call an arc-coloring of a digraph G the first type if it is an assignment of colors to the arc set of G in which no two consecutive arcs have the same color. In some researches, the arc-coloring of first type has been associated with the minimum number of the vertex-coloring called chromatic number. Considering the class of line digraphs, an arc-coloring of a digraph G of the first type is equivalent to the vertex-coloring of its line digraph L(G). In this paper, we study the arc-coloring of the first type and the vertex-coloring of line digraphs. We give the upper bound of the chromatic number of L(G) by the chromatic number of a digraph G which admits loops. It is also shown that there exists quite a small integer k so that the iterated line digraph Lk(G) is 3-vertex-colorable. As a consequence, we derive the chromatic number of de Bruijn and Kautz digraphs.

Consider a random digraph G=(V,A), where |V|=n and an arc (u,v) is present in A with probability p(n) independent of the existence of the other arcs. We discuss the expected number of vertices reachable from a vertex, the expected size of the transitive closure of G and their related topics based on the properties of reachability, where the reachability from a vertex s to t is defined as the probability that s is reachable to t. Let γn,p(n) denote the reachability s to t (s) in the above random digraph G. (In case of s=t, it requires another definition. ) We first present a method of computing the exact value of γn,p(n) for given n and p(n). Since the computation of γn,p(n) by this method requires O(n3) time, we then derive simple upper and lower bounds γn,p(n)U and γn,p(n)L on γn,p(n), respectively, and in addition, we give an upper bound n,p(n) on γn,p(n)U, which is easier to analyze but is still rather accurate. Then, we discuss the asymptotic behavior of n,p(n) and show that, if p(n)=α/(n-1), limnn,p(n) converges to one of the solutions of the equation 1-x-e-α x=0. Furthermore, as for (n) and (n), which are upper bounds on the expected number of reachable vertices and the expected size of the transitive closure of G, resp. , it turns out that limn(n) =α/(1-α) if p(n)=α/(n-1) for 0<α<1; otherwise either 0 or , and limn(n)=α if p(n)=α/(n-1)2 for α0; otherwise either 0 or .

Distributed web caching reduces retrieval latency of World Wide Web (WWW) objects such as text and graphics. Conventional distributed web caching methods, however, require many query messages among cache servers, which limits their scalability and reliability. To overcome these problems, we propose a query caching method in which each cache server caches not only WWW objects but also a query history. This method of finding cached objects can reduce the number of query messages among cache servers, making it possible to construct a large-scale distributed web cache server. We also propose an algorithm for constructing efficient query relationships among cache servers.

Topological sorting is, given with a directed acyclic graph G = (V, E), to find a total ordering of the vertices such that if (u, v) E then u is ordered before v. Instead of finding total orderings, we wish to find out how many total orderings exist in a given directed acyclic graph G = (V, E). Here we call a total ordering as legal sequence and the problem as legal sequence number problem. In this paper, we first propose theorems on equivalent transformation of graphs with respect to legal sequence number. Then we give a formula to calculate legal sequence number of basic series-parallel digraphs and a way of the calculation for general series-parallel digraphs. Finally we apply our results to show how to obtain legal sequence number for a class of extended series-parallel digraphs.