Strong geodetic problem on Cartesian products of graphs

The strong geodetic problem is a recent variation of the geodetic problem. For a graph $G$, its strong geodetic number ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that each vertex of $G$ lies on a fixed shortest path between a pair of vertices from $S$. In this paper, the strong geodetic problem is studied on the Cartesian product of graphs. A general upper bound for ${\rm sg}(G \,\square\, H)$ is determined, as well as exact values for $K_m \,\square\, K_n$, $K_{1, k} \,\square\, P_l$, and certain prisms. Connections between the strong geodetic number of a graph and its subgraphs are also discussed.Comment: 18 pages, 9 figure

The graph bottleneck identity

A matrix $S=(s_{ij})\in{\mathbb R}^{n\times n}$ is said to determine a \emph{transitional measure} for a digraph $G$ on $n$ vertices if for all $i,j,k\in\{1,\...,n\},$ the \emph{transition inequality} $s_{ij} s_{jk}\le s_{ik} s_{jj}$ holds and reduces to the equality (called the \emph{graph bottleneck identity}) if and only if every path in $G$ from $i$ to $k$ contains $j$. We show that every positive transitional measure produces a distance by means of a logarithmic transformation. Moreover, the resulting distance $d(\cdot,\cdot)$ is \emph{graph-geodetic}, that is, $d(i,j)+d(j,k)=d(i,k)$ holds if and only if every path in $G$ connecting $i$ and $k$ contains $j$. Five types of matrices that determine transitional measures for a digraph are considered, namely, the matrices of path weights, connection reliabilities, route weights, and the weights of in-forests and out-forests. The results obtained have undirected counterparts. In [P. Chebotarev, A class of graph-geodetic distances generalizing the shortest-path and the resistance distances, Discrete Appl. Math., URL http://dx.doi.org/10.1016/j.dam.2010.11.017] the present approach is used to fill the gap between the shortest path distance and the resistance distance.Comment: 12 pages, 18 references. Advances in Applied Mathematic

Statistical testing of directions observations independence

Independence of observations is often assumed when adjusting geodetic network. Unlike the\ud distance observations, no dependence of environmental conditions is known for horizontal\ud direction observations. In order to determine the dependence of horizontal direction observations,\ud we established test geodetic network of a station and four observation points. Measurements of\ud the highest possible accuracy were carried out using Leica TS30 total station along with precise\ud prisms GPH1P. Two series of hundred sets of angles were measured, with the first one in bad\ud observation conditions. Using different methods, i.e. variance–covariance matrices, x2 test and analyses of time series, the independence of measured directions, reduced directions and horizontal angles were tested. The results show that the independence of horizontal direction\ud observations is not obvious and certainly not in poor conditions. In this case, it would be appropriate for geodetic network adjustments to use variance–covariance matrix calculated from measurements instead of diagonal variance–covariance matrix

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