2-Point set domination in separable graphs

In a graph [Formula: see text], a set [Formula: see text] is a [Formula: see text]-point set dominating set (in short 2-psd set) of [Formula: see text] if for every subset [Formula: see text] there exists a nonempty subset [Formula: see text] containing at most two vertices such that the induced subgraph [Formula: see text] is connected in [Formula: see text]. The [Formula: see text]-point set domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of a 2-psd set of [Formula: see text]. The main focus of this paper is to find the value of [Formula: see text] for a separable graph and thereafter computing [Formula: see text] for some well-known classes of separable graphs. Further we classify the set of all 2-psd sets of a separable graph into six disjoint classes and study the existence of minimum 2-psd sets in each class.

Certificates of Factorisation for a Class of Triangle-Free Graphs

The chromatic polynomial $P(G,\lambda)$ gives the number of $\lambda$-colourings of a graph. If $P(G,\lambda)=P(H_{1},\lambda)P(H_{2},\lambda)/P(K_{r},\lambda)$, then the graph $G$ is said to have a chromatic factorisation with chromatic factors $H_{1}$ and $H_{2}$. It is known that the chromatic polynomial of any clique-separable graph has a chromatic factorisation. In this paper we construct an infinite family of graphs that have chromatic factorisations, but have chromatic polynomials that are not the chromatic polynomial of any clique-separable graph. A certificate of factorisation, that is, a sequence of rewritings based on identities for the chromatic polynomial, is given that explains the chromatic factorisations of graphs from this family. We show that the graphs in this infinite family are the only graphs that have a chromatic factorisation satisfying this certificate and having the odd cycle $C_{2n+1}$, $n\geq2$, as a chromatic factor.

Certificates of Factorisation for Chromatic Polynomials

The chromatic polynomial gives the number of proper $\lambda$-colourings of a graph $G$. This paper considers factorisation of the chromatic polynomial as a first step in an algebraic study of the roots of this polynomial. The chromatic polynomial of a graph is said to have a chromatic factorisation if $P({G},\lambda)=P({H_{1}},\lambda)P({H_{2}},\lambda)/P({K_{r}},\lambda)$ for some graphs $H_{1}$ and $H_{2}$ and clique $K_{r}$. It is known that the chromatic polynomial of any clique-separable graph, that is, a graph containing a separating $r$-clique, has a chromatic factorisation. We show that there exist other chromatic polynomials that have chromatic factorisations but are not the chromatic polynomial of any clique-separable graph and identify all such chromatic polynomials of degree at most 10. We introduce the notion of a certificate of factorisation, that is, a sequence of algebraic transformations based on identities for the chromatic polynomial that explains the factorisations for a graph. We find an upper bound of $n^{2}2^{n^{2}/2}$ for the lengths of these certificates, and find much smaller certificates for all chromatic factorisations of graphs of order $\leq 9$.

Emergent Features, Spatial Arithmetic, and Visual Scanning Behavior

In this study the relationship between spatial arithmetic and emergent features was investigated in a study requiring subjects to perform integrated tasks (tasks that require the integration of various data values shown in a graph) with one type of configural graph and one type of non-configural (separable) graph. Among other things, the question was addressed to what extent separable graphs have emergent features (perceptual qualities arising from the way the values are plotted) that can invoke spatial arithmetic (arithmetic using visual strategies) and thereby can facilitate task performance. To this end tasks were defined that were expected to invoke either spatial or mental (non-spatial) arithmetic. The subjects' visual scanning behavior was also recorded to see if it can supplement performance indices as a means for comparing spatial and mental arithmetic and for assessing the relative effectiveness by which graphical information is processed. The results of an experiment show that, as expected, spatial arithmetic can be invoked in both configural graphs and separable graphs. In addition, spatial arithmetic and mental arithmetic could be experimentally distinguished in terms of global characteristics of the visual scanning behavior. However, configural graphs did not result in better performance than separable graphs. These findings are discussed, the focus of the discussion being on the multitude of features that are often present in both configural and separable graphs and that allow for various degrees of spatial processing.

A Combinatorial Decomposition Theory

Given a finite undirected graphGandA⊆E(G),G(A)denotes the subgraph ofGhaving edge-setAand having no isolated vertices. For a partition {E1, E2}ofE(G),W(G; E1)denotes the setV(G(E1))⋂V(G(E2)). We say thatGisnon-separableif it is connected and for every proper, non-empty subsetAofE(G), we have |W(G;A)| ≧ 2. Asplitof a non-separable graphGis a partition {E1, E2} ofE(G)such that|E1| ≧ 2 ≧ |E2| and |W(G; E1)| = 2.Where {E1, E2} is a split ofG, W(G; E2)= {u, v}, andeis an element not inE(G),we form graphsGii= 1 and 2, by addingetoG(Ei)as an edge joiningutov.