1 Edge-disjoint decompositions Some of the graphs in this section are allowed to have multiple edges. They will be referred to as multigraphs. All graphs are assumed to have vertex set [n] = {1, 2, . . . , n}. Multiple edges between vertices of a multigraph G are regarded as distinct members of its edge set, E(G). We say that a collection of multigraphs1 Gi, i ∈ [m] is an edge-disjoint decompos...

one of the generalizations of the wiener number to weighted graphs is to assign probabilities to edges, meaning that in nonstatic conditions the edge is present only with some probability. the reliability wiener number is defined as the sum of reliabilities among pairs of vertices, where the reliability of a pair is the reliability of the most reliable path. closed expressions are derived for t...

Foucaud, Krishna and Ramasubramony Sulochana recently introduced the concept of monitoring edge-geodetic sets in graphs, a related graph invariant. These are vertices such that removal any edge changes distance between some pair set. They studied minimum possible size set given graph, which we call number. We show decision problem for number is NP-complete. also give best-possible upper lower b...

We study convexity properties of graphs. In this paper we present a linear-time algorithm for the geodetic number in tree-cographs. Settling a 10-year-old conjecture, we prove that the Steiner number is at least the geodetic number in AT-free graphs. Computing a maximal and proper monophonic set in AT-free graphs is NP-complete. We present polynomial algorithms for the monophonic number in perm...