Related papers: A characterization on orientations of graphs avoid…

A fourientation of a graph is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. Fixing a total order on the edges and a reference orientation of the graph, we…

A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not necessarily $(c:d)$-choosable. Applying probabilistic methods, an upper bound for the…

In this paper we study graphs which admit acyclic orientations that contain a pair of arc-disjoint out-branching and in-branching (such an orientation is called good) and we focus on edge-minimal such graphs. A 2T-graph is a graph whose…

Given a graph G=(V, E), a vertex is said to ve-dominate an edge if it is either incident with the edge or adjacent to one of its endpoints. A set of vertices is a ve-dominating set if it ve-dominates every edge of the graph. We introduce…

For a graph $G=(V(G),E(G))$, an Italian dominating function (ID function) $f:V(G)\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V(G)$ with $f(v)=0$, either $v$ is adjacent to a vertex assigned $2$ under $f$ or $v$ is…

Let $G$ be a graph. A good function is a function $f:V(G)\rightarrow \{-1,1\}$, satisfying $f(N(v))\geq 1$, for each $v\in V(G)$, where $ N(v)=\{u\in V(G)\, |\, uv\in E(G) \} $ and $f(S) = \sum_{u\in S} f(u)$ for every $S \subseteq V(G) $.…

Let $G=(V, E)$ be a simple and undirected graph. For some integer $k\geq 1$, a set $D\subseteq V$ is said to be a k-dominating set in $G$ if every vertex $v$ of $G$ outside $D$ has at least $k$ neighbors in $D$. Furthermore, for some real…

When facing graph signal processing tasks, the workhorse assumption is that the graph describing the support of the signals is known. However, in many relevant applications the available graph suffers from observation errors and…

We give a short, self-contained, and easily verifiable proof that determining the outerthickness of a general graph is NP-hard. This resolves a long-standing open problem on the computational complexity of outerthickness. Moreover, our…

A copy of a graph $F$ is called an $F$-copy. For any graph $G$, the $F$-isolation number of $G$, denoted by $\iota(G,F)$, is the size of a smallest subset $D$ of the vertex set of $G$ such that the closed neighbourhood $N[D]$ of $D$ in $G$…

For a graph $G=(V,E)$ of order $n$, a Roman $\{2\}$-dominating function $f:V\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V$ with $f(v)=0$, either $v$ is adjacent to a vertex assigned $2$ under $f$, or $v$ is adjacent…

An odd graceful labeling of a graph G=(V,E) is a function f:V(G)->[0,1,2,...,2|E(G)|-1} such that |f(u)-f(v)| is odd value less than or equal to 2|E(G)-1| for any u, v in V(G). In spite of the large number of papers published on the subject…

Being motivated by John Tantalo's Planarity Game, we consider straight line plane drawings of a planar graph $G$ with edge crossings and wonder how obfuscated such drawings can be. We define $obf(G)$, the obfuscation complexity of $G$, to…

We call a pair $(m,f)$ of integers, $m\geq 1$, $0\leq f \leq \binom{m}{2}$, \emph{absolutely avoidable} if there is $n_0$ such that for any pair of integers $(n,e)$ with $n>n_0$ and $0\leq e\leq \binom{n}{2}$ there is a graph on $n$…

For a graph $G$, let $odd(G)$ and $\omega(G)$ denote the number of odd components and the number of components of $G$, respectively. Then it is well-known that $G$ has a 1-factor if and only if $odd(G-S)\le |S|$ for all $S\subset V(G)$.…

We study an "above guarantee" version of the {\sc Longest Path} problem in directed graphs: We are given a graph $G$, two vertices $s$ and $t$ of $G$, and a non-negative integer $k$, and the objective is to determine whether $G$ contains a…

A leaf matching operation on a graph consists of removing a vertex of degree~$1$ together with its neighbour from the graph. For $k\geq 0$, let $G$ be a $d$-regular cyclically $(d-1+2k)$-edge-connected graph of even order. We prove that for…

We consider the problem of finding an edge in a hidden undirected graph $G = (V, E)$ with $n$ vertices, in a model where we only allowed queries that ask whether or not a subset of vertices contains an edge. We study the non-adaptive model…

Let $F$ be a $2$-regular graph of order $v$. The Oberwolfach problem $OP(F)$, posed in 1967 and still open, asks for a decomposition of $K_v$ into copies of $F$. In this paper we show that $OP(F)$ has a solution whenever $F$ has a…

Let G be an undirected simple graph having n vertices and let f be a function defined to be f:V(G) -> {0,..., n-1}. An f-factor of G is a spanning subgraph H such that degree of a vertex v in H is f(v) for every vertex v in V(G). The…