Let a graph G = (V;E) be given. In the path center problem we want to find a path P in G such that the maximum weighted distance of P to every vertex in V is minimized. In this paper a genetic algorithm and ahybrid of genetic and ant colony algorithms are presented for the path center problem. Some test problems are examined to compare the algorithms. The results show that for almost all exampl...

The dissociation number of a graph G is the number of vertices in a maximum size induced subgraph of G with vertex degree at most 1. A k-path vertex cover of a graph G is a subset S of vertices of G such that every path of order k in G contains at least one vertex from S. The minimum 3-path vertex cover is a dual problem to the dissociation number. For this problem we present an exact algorithm...

In routing, a source vertex sends a message to a target vertex and the routing algorithm decides a path from the source vertex to the target vertex. In 2013, Wang and Juan proposed a simple version of the WK-recursive pyramid networks, is called the basic WK-recursive pyramid network, which have received much attention recently. There are many literatures that study on this topology. This paper...

A bipartite graph G is bipanconnected if, for any two distinct vertices x and y of G, it contains an [x, y]-path of length l for each integer l satisfying dG(x, y) ≤ l ≤ |V (G)| − 1 and 2|(l − dG(x, y)), where dG(x, y) denotes the distance between vertices x and y in G and V (G) denotes the vertex set of G. We say a bipartite graph G is bipanpositionably bipanconnected if, for any two distinct ...

In this paper we show that the problem of finding a chordless path between a vertex s and a vertex t containing a vertex v remains NP-complete in bipartite graphs, thereby strengthening previous results on the same problem. We show a relation between this problem and two interval operators: the simple path interval operator in hypergraphs and the even-chorded path interval operator in graphs. W...

We say that a k-uniform hypergraph C is an `-cycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of adjacent edges (in the natural ordering of the edges) intersects in precisely ` vertices. We prove that if 1 ≤ ` < k and k − ` does not divide k then any k-uniform hypergraph on n vertices with minimum ...

We consider the question whether an infinite eulerian graph has a decomposition into circuits and rays if the graph has only finitely many, say n, vertices of infinite degree, and only finitely many finite components after the removal of the vertices of infinite degree. It is known that the answer is affirmative for n < 2 and negative for n > 4. We settle the remaining case n = 3, showing that ...

The graphs with no five-vertex induced path are still not understood. But in the triangle-free case, we can do this and one better; we give an explicit construction for all triangle-free graphs with no six-vertex induced path. Here are two examples: the 16-vertex Clebsch graph, and the graph obtained from a complete bipartite graph by subdividing a perfect matching. We show that every connected...