We are looking at a model of preferential attachment graphs that can evolve in five different ways. On any given iteration, a single vertex may be connected to the graph, a path may be connected to the graph, an edge may be added to the graph, an edge may be removed from a graph, or a vertex may be removed from a graph, with each option having a given probability of occurring. If a single verte...

A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. We address the complexity issue of the following problem, max rainbow matching: Given an edge-colored graph G, how large is the largest rainbow matching in G? We present several sharp contrasts in the complexity of this problem. We show, among others, that • max rainbow matching can be approximated by a ...

A quasi-hamiltonian path in a semicomplete multipartite digraph D is a path which visits each maximal independent set (also called a partite set) of D at least once. This is a generalization of a hamiltonian path in a tournament. In this paper we investigate the complexity of finding a quasi-hamiltonian path, in a given semicomplete multipartite digraph, from a prescribed vertex x to a prescrib...

A path partition of a graph is a collection of vertex-disjoint paths that cover all vertices of the graph. The path-partition problem is to find a path partition of minimum size. This paper gives a linear-time algorithm for the path-partition problem in bipartite distance-hereditary graphs.

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. Let G be a connected graph of order n with minimum degree at least two and with maximum degree at least three. We define a vertex as large if it has degree more than 2 and we let L be the...

A graph G is vertex pancyclic if for each vertex v ∈ V (G), and for each integer k with 3 ≤ k ≤ |V (G)|, G has a k-cycle Ck such that v ∈ V (Ck). Let s ≥ 0 be an integer. If the removal of at most s vertices in G results in a vertex pancyclic graph, we say G is an s-vertex pancyclic graph. Let G be a simple connected graph that is not a path, cycle or K1,3. Let l(G) = max{m : G has a divalent p...

1 I n t r o d u c t i o n All graphs in this paper are connected and simple, i.e. finite, undirected, loopless and without multiple edges. In a graph G = (V, E) the length of a path from a vertex v to a vertex u is the number of edges in the path. The distance d(u, v) from vertex u to vertex v is the length of a minimum length path from u to v and the interval I(u, v) between these vertices is ...

Leavitt path algebras are shown to be algebras of right quotients of their corresponding path algebras. Using this fact we obtain maximal algebras of right quotients from those (Leavitt) path algebras whose associated graph satisfies that every vertex connects to a line point (equivalently, the Leavitt path algebra has essential socle). We also introduce and characterize the algebraic counterpa...

A k-disjoint path cover of a graph is a set of k internally vertex-disjoint paths which cover the vertex set with k paths and each of which runs between a source and a sink. Given that each source and sink v is associated with an integer-valued demand d(v) ≥ 1, we are concerned with general-demand k-disjoint path cover in which every source and sink v is contained in the d(v) paths. In this pap...

Phylogenetic relationships may be represented by rooted acyclic directed graphs in which each vertex, corresponding to a taxon, possesses a genome. Assume the characters are all binary. A homoplasy occurs if a particular character changes its state more than once in the graph. A vertex is "regular" if it has only one parent and "hybrid" if it has more than one parent. A "regular path" is a dire...