The existence of a densely knit core surrounded by a loosely connected periphery is a common macro-structural feature of social networks. Formally, the CorePeriphery problem is to partition the nodes of an undirected graph G = (V,E) such that a subset X ⊂ V , the core, induces a dense subgraph, and its complement V \X, the periphery, induces a sparse subgraph. Split graphs represent the ideal c...

Many hard graph problems, such as Hamiltonian Cycle, become FPT when parameterized by treewidth, a parameter that is bounded only on sparse graphs. When parameterized by the more general parameter cliquewidth, Hamiltonian Cycle becomes W[1]-hard, as shown by Fomin et al. [5]. Sæther and Telle address this problem in their paper [13] by introducing a new parameter, split-matching-width, which li...

BSP trees and a priori potential-occlusion-graph-based techniques may be uniied to produce a superior algorithm for visible surface determination involving a hybrid data structure in which BSP nodes are used where the simple graph-based algorithm breaks down. In particular, empirical results show around two to three times fewer split polygons for this method, 20{40% less run-time space needed, ...

A bijection between split graphs and minimal covers of a set by subsets is presented. As the enumeration problem for such minimal covers has been solved, this implies that split graphs can also be enumerated. 1 Motivation A split graph is a chordal graph with a chordal complement. It is straightforward to recognize split graphs, and therefore to compute the numbers of split graphs on a small nu...

Electromagnetic topology was introduced a few decades ago in order to bring some rules in analysing EMC of very large systems through a fluence graph of electromagnetic interactions. Establishing these graphs was made possible from application of the so-called "good shielding approximation". At cable or track level, such a procedure is investigated in this paper. Other approximations are introd...

In this paper, given a split extension of an arbitrary Coxeter group by automorphisms of the Coxeter graph, we determine the involutions in that extension whose centralizer has finite index. Our result has applications to many problems such as the isomorphism problem of general Coxeter groups. In the argument, some properties of certain special elements and of the fixed-point subgroups by graph...

In this paper, we construct the Hall-Janko graph within the split Cayley hexagon H(4). Using this graph we then construct the near-octagon of order (2, 4) as a subgeometry of the dual of H(4), with J2 : 2 as its automorphism group. These constructions are based on a lemma determining the possibilities for the structure of the intersection of two subhexagons of order 2 in H(4).

Let G be a graph of order n with clique number ω(G), chromatic number χ(G) and independence number α(G). We show that χ(G) ≤ n+ω+1−α 2 . Moreover, χ(G) ≤ n+ω−α 2 , if either ω + α = n + 1 and G is not a split graph or α+ω = n−1 and G contains no induced Kω+3−C5.

In this paper, we investigate the well-studied Hamiltonian cycle problem, and present an interesting dichotomy result on split graphs. T. Akiyama, T. Nishizeki, and N. Saito [22] have shown that the Hamiltonian cycle problem is NP-complete in planar bipartite graph with maximum degree 3. Using this reduction, we show that the Hamiltonian cycle problem is NP-complete in split graphs. In particul...

A vertex ranking of a graph G is an assignment of positive integers (colors) to the vertices of G such that each path connecting two vertices of the same color contains a vertex of a higher color. Our main goal is to find a vertex ranking using as few colors as possible. Considering on-line algorithms for vertex ranking of split graphs, we prove that the worst case ratio of the number of colors...