Transit functions on graphs (and posets)

The notion of transit function is introduced to present a unifying approachfor results and ideas on intervals, convexities and betweenness in graphs andposets. Prime examples of such transit functions are the interval function I andthe induced path function J of a connected graph. Another transit function isthe all-paths function. New transit functions are introduced, such as the cutvertextransit function and the longest path function. The main idea of transitfunctions is that of ‘transferring’ problems and ideas of one transit functionto the other. For instance, a result on the interval function I might suggestsimilar problems for the induced path function J. Examples are given of howfruitful this transfer can be. A list of Prototype Problems and Questions forthis transferring process is given, which suggests many new questions and openproblems.graph theory;betweenness;block graph;convexity;distance in graphs;interval function;path function;induced path;paths and cycles;transit function;types of graphs

Convex Cycle Bases

Convex cycles play a role e.g. in the context of product graphs. We introduce convex cycle bases and describe a polynomial-time algorithm that recognizes whether a given graph has a convex cycle basis and provides an explicit construction in the positive case. Relations between convex cycles bases and other types of cycles bases are discussed. In particular we show that if G has a unique minimal cycle bases, this basis is convex. Furthermore, we characterize a class of graphs with convex cycles bases that includes partial cubes and hence median graphs. (authors' abstract)Series: Research Report Series / Department of Statistics and Mathematic

Alliance free sets in Cartesian product graphs

Let $G=(V,E)$ be a graph. For a non-empty subset of vertices $S\subseteq V$, and vertex $v\in V$, let $\delta_S(v)=|\{u\in S:uv\in E\}|$ denote the cardinality of the set of neighbors of $v$ in $S$, and let $\bar{S}=V-S$. Consider the following condition: {equation}\label{alliancecondition} \delta_S(v)\ge \delta_{\bar{S}}(v)+k, \{equation} which states that a vertex $v$ has at least $k$ more neighbors in $S$ than it has in $\bar{S}$. A set $S\subseteq V$ that satisfies Condition (\ref{alliancecondition}) for every vertex $v \in S$ is called a \emph{defensive} $k$-\emph{alliance}; for every vertex $v$ in the neighborhood of $S$ is called an \emph{offensive} $k$-\emph{alliance}. A subset of vertices $S\subseteq V$, is a \emph{powerful} $k$-\emph{alliance} if it is both a defensive $k$-alliance and an offensive $(k +2)$-alliance. Moreover, a subset $X\subset V$ is a defensive (an offensive or a powerful) $k$-alliance free set if $X$ does not contain any defensive (offensive or powerful, respectively) $k$-alliance. In this article we study the relationships between defensive (offensive, powerful) $k$-alliance free sets in Cartesian product graphs and defensive (offensive, powerful) $k$-alliance free sets in the factor graphs