To monitor an electric power system by placing as few phase measurement units (PMUs) as possible is closely related to the famous vertex cover problem and domination problem in graph theory. A set S is a power dominating set (PDS) of a graph G = (V,E), if every vertex and every edge in the system is observed following the observation rules of power system monitoring. The minimum cardinality of ...

The power system monitoring problem asks for as few as possible measurement devices to be put in an electric power system. The problem has a graph theory model involving power dominating sets in graphs. The power domination number γP (G) of G is the minimum cardinality of a power dominating set. Dorfling and Henning [2] determined the power domination number of the Cartesian product of paths. I...

Abstract We investigate the asymptotic number of induced subgraphs in power-law uniform random graphs. show that these appear typically on vertices with specific degrees, which are found by solving an optimization problem. Furthermore, we this problem allows to design a linear-time, randomized algorithm distinguishes between graphs and graph models create approximately desired degree sequence: ...

for any $k in mathbb{n}$, the $k$-subdivision of graph $g$ is a simple graph $g^{frac{1}{k}}$, which is constructed by replacing each edge of $g$ with a path of length $k$. in [moharram n. iradmusa, on colorings of graph fractional powers, discrete math., (310) 2010, no. 10-11, 1551-1556] the $m$th power of the $n$-subdivision of $g$ has been introduced as a fractional power of $g$, denoted by ...

We examine the simultaneous embedding with fixed edges problem for two planar graphs G1 and G2 with the focus on their intersection S = G1∩G2. In particular, we will present the complete set of intersection graphs S that guarantee a simultaneous embedding with fixed edges for (G1, G2). More formally, we define the subset ISEFE of all planar graphs as follows: A graph S lies in ISEFE if every pa...

We prove that the unique decomposition of connected graphs defined by Tutte is definable by formulas of Monadic Second-Order Logic. This decomposition has two levels: every connected graph is a tree of "2-connected components" called blocks ; every 2-connected graph is a tree of so-called 3-blocks. Our proof uses 2dags which are certain acyclic orientations of the considered graphs. We obtain a...

The rank-width is a graph parameter related in terms of fixed functions to cliquewidth but more tractable. Clique-width has nice algorithmic properties, but no good “minor” relation is known analogous to graph minor embedding for tree-width. In this paper, we discuss the vertex-minor relation of graphs and its connection with rank-width. We prove a relationship between vertex-minors of bipartit...

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Random graphs with given expected degrees G(w) were introduced by Chung and Lu so as to extend the theory of classical G(n, p) random graphs to include random power law graphs. We investigate asymptotic results for the game of Cops and Robber played on G(w) and G(n, p). Under mild conditions on the degree sequence w, an asymptotic lower bound for the cop number of G(w) is given. We prove that t...