A note on connected dominating sets of distance-hereditary graphs

A vertex subset of a graph is a dominating set if every vertex of the graph belongs to the set or has a neighbor in it. A connected dominating set is a dominating set such that the induced subgraph of the set is a connected graph. A graph is called distance-hereditary if every induced path is a shortest path. In this note, we give a complete description of the (inclusionwise) minimal connected dominating sets of connected distance-hereditary graphs in the following sense: If G is a connected distance-hereditary graph that has a dominating vertex, any minimal connected dominating set is a single vertex or a pair of two adjacent vertices. If G does not have a dominating vertex, the subgraphs induced by any two minimal connected dominating sets are isomorphic. In particular, any inclusionwise minimal connected dominating set of a connected distance-hereditary graph without dominating vertex has minimal size. In other words, connected distance-hereditary graphs without dominating vertex are connected well-dominated. Furthermore, we show that if G is a distance-hereditary graph that has a minimal connected dominating set X of size at least 2, then for any connected induced subgraph H it holds that the subgraph induced by any minimal connected dominating set of H is isomorphic to an induced subgraph of G[X]

A note on connected dominating sets of distance-hereditary graphs

A vertex subset of a graph is a dominating set if every vertex of the graph belongs to the set or has a neighbor in it. A connected dominating set is a dominating set such that the induced subgraph of the set is a connected graph. A graph is called distance-hereditary if every induced path is a shortest path. In this note, we give a complete description of the (inclusionwise) minimal connected dominating sets of connected distance-hereditary graphs in the following sense: If G is a connected distance-hereditary graph that has a dominating vertex, any minimal connected dominating set is a single vertex or a pair of two adjacent vertices. If G does not have a dominating vertex, the subgraphs induced by any two minimal connected dominating sets are isomorphic. In particular, any inclusionwise minimal connected dominating set of a connected distance-hereditary graph without dominating vertex has minimal size. In other words, connected distance-hereditary graphs without dominating vertex are connected well-dominated. Furthermore, we show that if G is a distance-hereditary graph that has a minimal connected dominating set X of size at least 2, then for any connected induced subgraph H it holds that the subgraph induced by any minimal connected dominating set of H is isomorphic to an induced subgraph of GX

Steiner Distance in Product Networks

For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n$ and $k$ be two integers with $2\leq k\leq n$. Then the \emph{Steiner $k$-eccentricity $e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=\max \{d_G(S)\,|\,S\subseteq V(G), \ |S|=k, \ and \ v\in S\}$. Furthermore, the \emph{Steiner $k$-diameter} of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\, v\in V(G)\}$. In this paper, we investigate the Steiner distance and Steiner $k$-diameter of Cartesian and lexicographical product graphs. Also, we study the Steiner $k$-diameter of some networks.Comment: 29 pages, 4 figure

Networks with small stretch number

Abstract In a previous work, the authors introduced the class of graphs with bounded induced distance of order k (BID(k) for short), to model non-reliable interconnection networks. A network modeled as a graph in BID(k) can be characterized as follows: if some nodes have failed, as long as two nodes remain connected, the distance between these nodes in the faulty graph is at most k times the distance in the non-faulty graph. The smallest k such that G∈BID(k) is called stretch number of G. We show an odd characteristic of the stretch numbers: every rational number greater or equal 2 is a stretch number, but only discrete values are admissible for smaller stretch numbers. Moreover, we give a new characterization of classes BID(2−1/i), i⩾1, based on forbidden induced subgraphs. By using this characterization, we provide a polynomial time recognition algorithm for graphs belonging to these classes, while the general recognition problem is Co-NP-complete

Graph Algorithms and Applications

The mixture of data in real-life exhibits structure or connection property in nature. Typical data include biological data, communication network data, image data, etc. Graphs provide a natural way to represent and analyze these types of data and their relationships. Unfortunately, the related algorithms usually suffer from high computational complexity, since some of these problems are NP-hard. Therefore, in recent years, many graph models and optimization algorithms have been proposed to achieve a better balance between efficacy and efficiency. This book contains some papers reporting recent achievements regarding graph models, algorithms, and applications to problems in the real world, with some focus on optimization and computational complexity