We prove that every internally 4-connected non-planar bipartite graph has an odd K3,3 subdivision; that is, a subgraph obtained from K3,3 by replacing its edges by internally disjoint odd paths with the same ends. The proof gives rise to a polynomial-time algorithm to find such a subdivision. (A bipartite graph G is internally 4-connected if it is 3-connected, has at least five vertices, and th...

Recently, the subdivision Randic index was introduced. In this paper, we present new version of Randic index by using some graph operator and in related to the subdivision Randic index. Next, by using some results about this version, it is computed for TUC C (R) nanotubes. 4 8

Shary considers the method “to turn out better than the traditional techniques from [11, 6, 8] in either the computational efficacy and the quality of the results it produces”. Although Shary presents it as an isolated phenomenon, it turns out that the graph subdivision method is intimately intertwined with a variety of methods in optimization. It is the purpose of this paper to present a unifi...

A graph G is said to have property Pm if it contains no subdivision of Km+1 and no subdivision of Kdm=2e+1;bm=2c+1. Chartrand et al. (J. Combin Theory 10 (1971) 12–41) (see also Problem 6.3 in Jensen and Toft (Graph Coloring Problems, Wiley, New York, 1995) conjectured that the set of vertices (respectively, edges) of any graph with property Pm can be partitioned into m−n+1 subsets such that ea...

There is a belief that subdivision schemes require the subdominant eigenvalue, λ , to be the same around extraordinary vertices as in the regular regions of the mesh [Barthe and Kobbelt, 2004, Zulti et al., 2006, Ni and Nasri, 2006]. This belief is owing to the polar rendering artifacts which occur around extraordinary points when λ is significantly larger than in the regular regions [Sabin and...

A set S of vertices in a graph G = (V,E) is a total dominating set of G if every vertex of V is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set of G. The total domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the tot...

A set D of vertices in a graph G = (V, E) is a weakly connected dominating set of G if D is dominating in G and the subgraph weakly induced by D is connected. The weakly connected domination number of G is the minimum cardinality of a weakly connected dominating set of G. The weakly connected domination subdivision number of a connected graph G is the minimum number of edges that must be subdiv...

Although previous studies have attempted to address the challenge of land subdivision, there is still a scarcity in the literature on how the extent of conformity to land subdivision planning regulations may be statistically and spatially analyzed. This study, therefore, explores the extent to which residential developers comply with land subdivision planning regulations in Kenya, a case study ...