Diameter of orientations of graphs with given order and number of blocks

A strong orientation of a graph $G$ is an assignment of a direction to each edge such that $G$ is strongly connected. The oriented diameter of $G$ is the smallest diameter among all strong orientations of $G$. A block of $G$ is a maximal connected subgraph of $G$ that has no cut vertex. A block graph is a graph in which every block is a clique. We show that every bridgeless graph of order $n$ containing $p$ blocks has an oriented diameter of at most $n-\lfloor \frac{p}{2} \rfloor$. This bound is sharp for all $n$ and $p$ with $p \geq 2$. As a corollary, we obtain a sharp upper bound on the oriented diameter in terms of order and number of cut vertices. We also show that the oriented diameter of a bridgeless block graph of order $n$ is bounded above by $\lfloor \frac{3n}{4} \rfloor$ if $n$ is even and $\lfloor \frac{3(n+1)}{4} \rfloor$ if $n$ is odd.Comment: 15 pages, 2 figure

Algorithmic aspects of disjunctive domination in graphs

For a graph $G=(V,E)$, a set $D\subseteq V$ is called a \emph{disjunctive dominating set} of $G$ if for every vertex $v\in V\setminus D$, $v$ is either adjacent to a vertex of $D$ or has at least two vertices in $D$ at distance $2$ from it. The cardinality of a minimum disjunctive dominating set of $G$ is called the \emph{disjunctive domination number} of graph $G$, and is denoted by $\gamma_{2}^{d}(G)$. The \textsc{Minimum Disjunctive Domination Problem} (MDDP) is to find a disjunctive dominating set of cardinality $\gamma_{2}^{d}(G)$. Given a positive integer $k$ and a graph $G$, the \textsc{Disjunctive Domination Decision Problem} (DDDP) is to decide whether $G$ has a disjunctive dominating set of cardinality at most $k$. In this article, we first propose a linear time algorithm for MDDP in proper interval graphs. Next we tighten the NP-completeness of DDDP by showing that it remains NP-complete even in chordal graphs. We also propose a $(\ln(\Delta^{2}+\Delta+2)+1)$-approximation algorithm for MDDP in general graphs and prove that MDDP can not be approximated within $(1-\epsilon) \ln(|V|)$ for any $\epsilon>0$ unless NP $\subseteq$ DTIME$(|V|^{O(\log \log |V|)})$. Finally, we show that MDDP is APX-complete for bipartite graphs with maximum degree $3$

Dynamic Changes in Protein Functional Linkage Networks Revealed by Integration with Gene Expression Data

Response of cells to changing environmental conditions is governed by the dynamics of intricate biomolecular interactions. It may be reasonable to assume, proteins being the dominant macromolecules that carry out routine cellular functions, that understanding the dynamics of protein∶protein interactions might yield useful insights into the cellular responses. The large-scale protein interaction data sets are, however, unable to capture the changes in the profile of protein∶protein interactions. In order to understand how these interactions change dynamically, we have constructed conditional protein linkages for Escherichia coli by integrating functional linkages and gene expression information. As a case study, we have chosen to analyze UV exposure in wild-type and SOS deficient E. coli at 20 minutes post irradiation. The conditional networks exhibit similar topological properties. Although the global topological properties of the networks are similar, many subtle local changes are observed, which are suggestive of the cellular response to the perturbations. Some such changes correspond to differences in the path lengths among the nodes of carbohydrate metabolism correlating with its loss in efficiency in the UV treated cells. Similarly, expression of hubs under unique conditions reflects the importance of these genes. Various centrality measures applied to the networks indicate increased importance for replication, repair, and other stress proteins for the cells under UV treatment, as anticipated. We thus propose a novel approach for studying an organism at the systems level by integrating genome-wide functional linkages and the gene expression data

Average distance and independence number

AbstractA sharp upper bound on the average distance of a graph depending on the order and the independence number is given. As a corollary we obtain the maximum average distance of a graph with given order and matching number. All extremal graphs are determined

Embedding graphs as isometric medians

AbstractWe show that every connected graph can be isometrically (i.e., as a distance preserving subgraph) embedded in some connected graph as its median. As an auxiliary result we also show that every connected graph is an isometric subgraph of some Cayley graph