Related papers: (k-2)-linear connected components in hypergraphs o…
In the Maker-Breaker positional game, Maker and Breaker take turns picking vertices of a hypergraph $H$, and Maker wins if and only if she possesses all the vertices of some edge of $H$. Deciding the outcome (i.e. which player has a winning…
The well-known Disjoint Paths problem is to decide if a graph contains k pairwise disjoint paths, each connecting a different terminal pair from a set of k distinct pairs. We determine, with an exception of two cases, the complexity of the…
A Hamiltonian path (a Hamiltonian cycle) in a graph is a path (a cycle, respectively) that traverses all of its vertices. The problems of deciding their existence in an input graph are well-known to be NP-complete, in fact, they belong to…
We prove that a connected graph has linear rank-width 1 if and only if it is a distance-hereditary graph and its split decomposition tree is a path. An immediate consequence is that one can decide in linear time whether a graph has linear…
Testing whether there is an induced path in a graph spanning k given vertices is already NP-complete in general graphs when k=3. We show how to solve this problem in polynomial time on claw-free graphs, when k is not part of the input but…
A temporal (directed) graph is a graph whose edges are available only at specific times during its lifetime, $\tau$. Paths are sequences of adjacent edges whose appearing times are either strictly increasing or non-strictly increasingly…
In a graph whose vertices are assigned integer ranks, a path is well-ranked if the endpoints have distinct ranks or some interior point has a higher rank than the endpoints. A ranking is an assignment of ranks such that all nontrivial paths…
It is proven that the connected pathwidth of any graph $G$ is at most $2\cdot\pw(G)+1$, where $\pw(G)$ is the pathwidth of $G$. The method is constructive, i.e. it yields an efficient algorithm that for a given path decomposition of width…
We consider high-order connectivity in $k$-uniform hypergraphs defined as follows: Two $j$-sets are $j$-connected if there is a walk of edges between them such that two consecutive edges intersect in at least $j$ vertices. We describe the…
We define an algorithm k which takes a connected graph G on a totally ordered vertex set and returns an increasing tree R (which is not necessarily a subtree of G). We characterize the set of graphs G such that k(G)=R. Because this set has…
A perfect matching in a hypergraph is a set of edges that partition the set of vertices. We study the complexity of deciding the existence of a perfect matching in orderable and separable hypergraphs. We show that the class of orderable…
Linear rank-width is a linearized variation of rank-width, and it is deeply related to matroid path-width. In this paper, we show that the linear rank-width of every $n$-vertex distance-hereditary graph, equivalently a graph of rank-width…
We call a digraph {\em $h$-semicomplete} if each vertex of the digraph has at most $h$ non-neighbors, where a non-neighbor of a vertex $v$ is a vertex $u \neq v$ such that there is no edge between $u$ and $v$ in either direction. This…
In this paper we define a construct called a time-graph. A complete time-graph of order n is the cartesian product of a complete graph with n vertices and a linear graph with n vertices. A time-graph of order n is given by a subset of the…
For a connected graph, a path containing all vertices is known as \emph{Hamiltonian path}. For general graphs, there is no known necessary and sufficient condition for the existence of Hamiltonian paths and the complexity of finding a…
We give a complete classification of the Kronecker (i.e. direct) product graphs that are planar and $3$-connected (i.e. $3$-polytopal). They are all of the form \[H\wedge K_2,\] where $H$ is a $2$-connected graph, possibly non-planar, and…
In this paper, we investigate Hamiltonian path problem in the context of split graphs, and produce a dichotomy result on the complexity of the problem. Our main result is a deep investigation of the structure of $K_{1,4}$-free split graphs…
A Hamiltonian path (cycle) in a graph is a path (cycle, respectively) which passes through all of its vertices. The problems of deciding the existence of a Hamiltonian cycle (path) in an input graph are well known to be NP-complete, and…
The \textit{longest path transversal number} of a connected graph $G$, denoted by $lpt(G)$, is the minimum size of a set of vertices of $G$ that intersects all longest paths in $G$. We present constant upper bounds for the longest path…
A path in an edge-colored graph is said to be rainbow if no color repeats on it. An edge-colored graph is said to be rainbow $k$-connected if every pair of vertices is connected by $k$ internally disjoint rainbow paths. The rainbow…