We show that planar embeddable 3-connected Laman graphs are generically non-soluble. A Laman graph represents a configuration of points on the Euclidean plane with just enough distance specifications between them to ensure rigidity. Formally, a Laman graph is a maximally independent graph, that is, one that satisfies the vertex-edge count 2v − 3 = e together with a corresponding inequality for ...

We show that planar embeddable 3-connected CAD graphs are generically non-soluble. A CAD graph represents a configuration of points on the Euclidean plane with just enough distance dimensions between them to ensure rigidity. Formally, a CAD graph is a maximally independent graph, that is, one that satisfies the vertex-edge count 2v − 3 = e together with a corresponding inequality for each subgr...

A connected graph with edge connectivity λ and minimum degree δ is called maximally edge-connected if λ = δ. A strongly connected digraph with arc connectivity λ and minimum degree δ is called maximally arc-connected if λ = δ. In this paper, some new sufficient conditions are presented for maximally edge-connected graphs and arc-connected digraphs. We only consider finite graphs (digraphs) with...

The edge-connectivity of a graph G can be defined as λ(G) = min{λG(u, v) |u, v ∈ V (G)}, where λG(u, v) is the local edge-connectivity of two vertices u and v in G. We call a graph G maximally edgeconnected when λ(G) = δ(G) and maximally local edge-connected when λG(u, v) = min{d(u), d(v)} for all pairs u and v of distinct vertices in G. In 2000, Fricke, Oellermann and Swart (unpublished manusc...

In this work, first, we present sufficient conditions for a bipartite digraph to attain optimum values of a stronger measure of connectivity, the so-called superconnectivity. To be more precise, we study the problem of disconnecting a maximally connected bipartite (di)graph by removing nontrivial subsets of vertices or edges. Within this framework, both an upper-bound on the diameter and Chartr...

Let G=(V,E) be a simple connected graph of order p and size q. A decomposition of a graph G is a collection π of edge-disjoint subgraphs G_1,G_2,…,G_n of G such that every edge of G belongs to exactly one G_i,(1≤i ≤n). The decomposition 〖π={G〗_1,G_2,…,G_n} of a connected graph G is said to be a distinct edge geodetic decomposition if g_1 (G_i )≠g_1 (G_j ),(1≤i≠j≤n). The maximum cardinality of π...

A connected digraph of minimum degree δ is said to be maximally connected if the cardinality of every cutset is al least δ, and it is called superconnected if moreover every minimum disconnecting set F consists of the vertices adjacent to or from a given vertex not belonging to F . Let π be an integer, 0 ≤ π ≤ δ − 2, δ being the minimum degree of the digraph. In this work, we prove that if D ≤ ...