Related papers: Listing Small Minimal $s,t$-separators in FPT-Dela…
Let ${\cal G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of ${\cal G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to ${\cal G}$. We denote by ${\cal A}_k ({\cal G})$ the set…
An isolating set of a graph is a set of vertices $S$ such that, if $S$ and its neighborhood is removed, only isolated vertices remain; and the isolation number is the minimum size of such a set. It is known that for every connected graph…
Let $G$ be a directed planar graph of complexity $n$, each arc having a nonnegative length. Let $s$ and $t$ be two distinct faces of $G$; let $s_1,...,s_k$ be vertices incident with $s$; let $t_1,...,t_k$ be vertices incident with $t$. We…
A set $D$ of vertices of a graph is a \emph{defensive alliance} if, for each element of $D$, the majority of its neighbours are in $D$. We consider the notion of local minimality in this paper. We are interested in finding a locally minimal…
A target set selection model is a graph $G$ with a threshold function $\tau:V\to \mathbb{N}$ upper-bounded by the vertex degree. For a given model, a set $S_0\subseteq V(G)$ is a target set if $V(G)$ can be partitioned into non-empty…
A commonly studied means of parameterizing graph problems is the deletion distance from triviality (Guo et al. 2004), which counts vertices that need to be deleted from a graph to place it in some class for which efficient algorithms are…
The planar separator theorem by Lipton and Tarjan [FOCS '77, SIAM Journal on Applied Mathematics '79] states that any planar graph with $n$ vertices has a balanced separator of size $O(\sqrt{n})$ that can be found in linear time. This…
A popular way to define or characterize graph classes is via forbidden subgraphs or forbidden minors. These characterizations play a key role in graph theory, but they rarely lead to efficient algorithms to recognize these classes. In…
A graph G is distinguished if its vertices are labelled by a map \phi: V(G) \longrightarrow {1,2,...,k} so that no graph automorphism preserves \phi. The distinguishing number of G is the minimum number k necessary for \phi to distinguish…
In the Disjoint Shortest Paths problem one is given a graph $G$ and a set $\mathcal{T}=\{(s_1,t_1),\dots,(s_k,t_k)\}$ of $k$ vertex pairs. The question is whether there exist vertex-disjoint paths $P_1,\dots,P_k$ in $G$ so that each $P_i$…
A set $S$ of vertices of a graph is a defensive alliance if, for each element of $S$, the majority of its neighbours is in $S$. We consider the notion of local minimality in this paper. We are interested in locally minimal defensive…
The paper presents fault-tolerant (FT) labeling schemes for general graphs, as well as, improved FT routing schemes. For a given $n$-vertex graph $G$ and a bound $f$ on the number of faults, an $f$-FT connectivity labeling scheme is a…
The minimum graph cut and minimum $s$-$t$-cut problems are important primitives in the modeling of combinatorial problems in computer science, including in computer vision and machine learning. Some of the most efficient algorithms for…
The Target Set Selection problem takes as an input a graph $G$ and a non-negative integer threshold $ \mbox{thr}(v) $ for every vertex $v$. A vertex $v$ can get active as soon as at least $ \mbox{thr}(v) $ of its neighbors have been…
We show that there is a deterministic local algorithm (constant-time distributed graph algorithm) that finds a 5-approximation of a minimum dominating set on outerplanar graphs. We show there is no such algorithm that finds a…
Algorithms for listing the subgraphs satisfying a given property (e.g.,being a clique, a cut, a cycle, etc.) fall within the general framework of set systems. A set system (U, F) uses a ground set U (e.g., the network nodes) and an…
In this paper, we present FPT-algorithms for special cases of the shortest vector problem (SVP) and the integer linear programming problem (ILP), when matrices included to the problems' formulations are near square. The main parameter is…
Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As…
In this article, we study connections between components of the Cayley graph $\mathrm{Cay}(G,A)$, where $A$ is an arbitrary subset of a group $G$, and cosets of the subgroup of $G$ generated by $A$. In particular, we show how to construct…
Partitioning a graph into balanced components is important for several applications. For multi-objective problems, it is useful not only to find one solution but also to enumerate all the solutions with good values of objectives. However,…