A NOTE ON USELESS COMMUNICATION CHANNELS AND DIRECTED GRAPHS

A note on vague graphs

In this paper, we introduce the notions of product vague graph, balanced product vague graph, irregularity and total irregularity of any irregular vague graphs and some results are presented. Also, density and balanced irregular vague graphs are discussed and some of their properties are established. Finally we give an application of vague digraphs.

A note on mixed graphs and directed splitting off

We give counterexamples to two conjectures in \Bill Jackson, Some remarks on arc-connectivity, vertex splitting, and orientation in graphs and digraphs, Journal of Graph Theory, 12(3):429{436, 1988" concerning orientations of mixed graphs and splitting oo in digraphs, and prove the rst conjecture in the (di-) Eulerian case(s). Beside that we solve the degree constrained directed augmentation pr...

Finding All Useless Arcs in Directed Planar Graphs

Maximum flow is a fundamental problem in Combinatorial Optimization that has numerous applications in both theory and practice. In this paper, we study the flow network simplification problem, which asks to remove all the useless arcs from the graph. To be precise, an arc is useless if it does not participate in any simple s, t-path. Weihe [FOCS’94, JCSS’97] showed if there exists an O(n log n)...

A Note on A Family of Directed Strongly Regular Graphs

where J is the all ones matrix and I is the identity. These matrix conditions are equivalent to the combinatorial conditions that the graph is both inand out-regular, and that the number of directed 2-paths from a vertex x to a vertex y is t if x = y, λ if x → y, and μ otherwise. Recently, these graphs were studied by Klin et al. [2], including some new constructions and a list of feasible para...

A Note on Strongly Quotient Graphs