It is NP-hard to determine the Radon number of graphs in the geodetic convexity. However, for certain classes of graphs, this well-known convexity parameter can be determined efficiently. In this paper, we focus on geodetic convexity spaces built upon d-dimensional grids, which are the Cartesian products of d paths. After revisiting a result of Eckhoff concerning the Radon number of Rd in the c...

We conjecture that, for each tree T , there exists a natural number kT such that the following holds: If G is a kT -edge-connected graph such that |E(T )| divides |E(G)|, then the edges of G can be divided into parts, each of which is isomorphic to T . We prove that for T = K1,3 (the claw), this holds if and only if there exists a (smallest) natural number kt such that every kt-edge-connected g...

For two vertices u and v of a connected graph G, the set IG[u, v] consists of all those vertices lying on u − v geodesics in G. Given a set S of vertices of G, the union of all sets IG[u, v] for u, v ∈ S is denoted by IG[S]. A set S ⊆ V (G) is a geodetic set if IG[S] = V (G) and the minimum cardinality of a geodetic set is its geodetic number g(G) of G. Bounds for the geodetic number of strong ...

Given two r-graphs G and H, an H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H. The minimum number of parts in an H-decomposition of G is denoted by φH(G). By a 2-edge-decomposition of an r-graph we mean an H-decomposition for any fixed r-graph H with exactly 2 edges. In the special case where the two edges of...

In this paper, we discuss the vertex space domain decomposition method (VSDDM) for solving the algebraic system of equations which arise from the discretization of symmetric and positive de nite elliptic boundary value problems via nite element methods on general unstructured meshes. Our theory does not require that the subdomains are shape regular and coarse grid is nested to the ne grid. Furt...