The domination game is played on an arbitrary graph $G$ by two players, Dominator and Staller. It is known that verifying whether the game domination number of a graph is bounded by a given integer $k$ is PSPACE-complete. On the other hand, it is showed in this paper that the problem can be solved for a graph $G$ in $mathcal O(Delta(G)cdot |V(G)|^k)$ time. In the special case when $k=3$ and the...

By definition, a P2-graph Γ is an undirected graph in which every vertex is contained in a path of length two. For such a graph, pc(Γ) denotes the minimum number of paths of length two that cover all n vertices of Γ. We prove that ⌈n/3⌉ ≤ pc(Γ) ≤ ⌊n/2⌋ and show that these upper and lower bounds are tight. Furthermore we show that every connected P2-graph Γ contains a spanning tree T such that p...

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 ...

This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects. It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014. 1 Covers by paths and cycles In this survey r-coloring always means edge-colori...

We show that the problem of deciding whether a digraph has a Hamiltonian path between two specified vertices and the problem of deciding whether a given graph has a cubic subgraph are complete for monotone NP via monotone projection translations. We also show that the problem of deciding whether a uniquely partially orderable (resp. comparability) graph has a cubic subgraph is complete for NP v...

We study first passage percolation on the random graph Gp(N) with exponentially distributed weights on the links. For the special case of the complete graph this problem can be described in terms of a continuous time Markov chain and recursive trees. The Markov chain X(t) describes the number of nodes that can be reached from the initial node in time t. The recursive trees, which are uniform tr...

We prove a number of results around kernelization of problems parameterized by the size of a given vertex cover of the input graph. We provide three sets of simple general conditions characterizing problems admitting kernels of polynomial size. Our characterizations not only give generic explanations for the existence of many known polynomial kernels for problems like q-Coloring, Odd Cycle Tran...

In this paper, we show that given a weighted, directed planar graph G, and any > 0, there exists a polynomial time and O(n 1 2+ ) space algorithm that computes the shortest path between two fixed vertices in G. We also consider the RedBluePath problem, which states that given a graph G whose edges are colored either red or blue and two fixed vertices s and t in G, is there a path from s to t in...

RDF is a knowledge representation language dedicated to the annotation of resourceswithin the framework of the semantic web. Among the query languages for RDF, SPARQLallows querying RDF through graph patterns, i.e., RDF graphs involving variables. Otherlanguages, inspired by the work in databases, use regular expressions for searching pathsin RDF graphs. Each approach can expres...