Let G = (V, E) be a graph or digraph and r : V → Z+. An r-detachment of G is a graph H obtained by ‘splitting’ each vertex v ∈ V into r(v) vertices. The vertices v1, ..., vr(v) obtained by splitting v are called the pieces of v in H. Every edge uv ∈ E corresponds to an edge of H connecting some piece of u to some piece of v. Crispin Nash-Williams [9] gave necessary and sufficient conditions for...

by JIANPING ZHU (Under the Direction of Robert W. Robinson) ABSTRACT In this thesis, we consider the following problem: Given a strongly connected digraph G = (V, E), where V is the set of vertices and E is the set of edges , “is it a minimal strong connected digraph?”. A reducible edges e is one for which G−e is strongly connected. A minimal strongly connected digraph is one with no reducible ...

We introduce a model for NP-hard problems pertaining to the connectivity of graphs, and design approximation algorithms for some of the key problems in this model. One of the well-known NP-hard problems is to find a minimum-cost strongly connected spanning subgraph of a directed network. In the minimum-cost k-(S, T ) connected digraph (abbreviated, k-(S, T ) connectivity) problem we are given a...

Competition graphs were rst introduced by Joel Cohen in the study of food webs and have since been extensively studied. Graphs which are the competition graph of a strongly connected or Hamiltonian digraph are of particular interest in applications to communication networks. It has been previously established that every graph without isolated vertices (except K 2) which is the competitiongraph ...

In this dissertation, we study the following two topics, i.e., the union closed set conjecture and the maximum edges cut in connected digraphs. The union-closed-set-conjecture-topic goes as follows. A finite family of finite sets is union closed if it contains the union of any two sets in it. Let XF = ∪F∈FF . A union closed family of sets is separating if for any two distinct elements in F , th...

We consider monotonicity problems for graph searching games. Variants of these games – defined by the type of moves allowed for the players – have been found to be closely connected to graph decompositions and associated width measures such as pathor tree-width. Of particular interest is the question whether these games are monotone, i.e. whether the cops can catch a robber without ever allowin...

We show that the edges of any graph $G$ containing two edge-disjoint spanning trees can be blue/red coloured so blue and red graphs are connected degrees at each vertex differ by most four. This improves a result H\"orsch. discuss variations question for digraphs, infinite computational question, resolve further questions H\"orsch in negative.

Bollobás and Scott proved that if the weighted outdegree of every vertex of a weighted digraph is at least 1, then it contains a (directed) path of weight at least 1. In this note we characterize the extremal weighted digraphs with no heavy paths. Our result is an extension of a corresponding theorem of Bondy and Fan in weighted graphs. Besides, we give examples showing the invalidity of genera...

We discuss relations between several known (some false, some open) conjectures on 3-edgeconnected, cubic graphs and how they all fit into the same framework related to cuts in matchings. We then provide a construction of 3-edge-connected digraphs satisfying the property that for every even subgraph E, the graph obtained by contracting the edges of E is not strongly connected. This disproves a r...

In 1980 Piere Duchet conjectured that odd directed cycles are the only edge minimal kernel-less connected digraphs i.e. in which after the removal of any edge a kernel appears. Although this conjecture was disproved recently by Apartsin, Ferapontova and Gurvich (1996), the following modiication of Duchet's conjecture still holds: odd holes (i.e. odd non-directed chordless cycles of length 5 or ...