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We consider the problem of decomposing the edges of a directed graph into as few paths as possible. There is a natural lower bound for the number of paths needed in an edge decomposition of a directed graph $D$ in terms of its degree…
Graph transformation is the rule-based modification of graphs, and is a discipline dating back to the 1970s. In general, to match the left-hand graph of a fixed rule within a host graph requires polynomial time, but to improve matching…
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…
The complete symmetric directed graph of order $v$, denoted $K_{v}^*$, is the directed graph on $v$ vertices that contains both arcs $(x,y)$ and $(y,x)$ for each pair of distinct vertices $x$ and $y$. For a given directed graph, $D$, the…
Let $G$ be a group. The directed endomorphism graph, $\dend(G)$ of $G$ is a directed graph with vertex set $G$ and there is a directed edge from the vertex $a$ to the vertex $b$ if $a \neq b$ and there exists an endomorphism on $G$ mapping…
An undirected graph is Eulerian if it is connected and all its vertices are of even degree. Similarly, a directed graph is Eulerian, if for each vertex its in-degree is equal to its out-degree. It is well known that Eulerian graphs can be…
We introduce a new arc in directed graphs of integers. Among other things, we determine the positive integers that have arcs to all except a finite number of positive integers. We also propose some possible research problems at the end of…
We consider the worst-case query complexity of some variants of certain \cl{PPAD}-complete search problems. Suppose we are given a graph $G$ and a vertex $s \in V(G)$. We denote the directed graph obtained from $G$ by directing all edges in…
There are several notions of gonality for graphs. The divisorial gonality dgon(G) of a graph G is the smallest degree of a divisor of positive rank in the sense of Baker-Norine. The stable gonality sgon(G) of a graph G is the minimum degree…
An out-tree $T$ of a directed graph $D$ is a rooted tree subgraph with all arcs directed outwards from the root. An out-branching is a spanning out-tree. By $l(D)$ and $l_s(D)$ we denote the maximum number of leaves over all out-trees and…
A graph $G$ is a $k$-leaf power if there exists a tree $T$ whose leaf set is $V(G)$, and such that $uv \in E(G)$ if and only if the distance between $u$ and $v$ in $T$ is at most $k$. The graph classes of $k$-leaf powers have several…
Graph transformation is the rule-based modification of graphs, and is a discipline dating back to the 1970s. The declarative nature of graph rewriting rules comes at a cost. In general, to match the left-hand graph of a fixed rule within a…
The directed power graph $\mathcal G(\mathbf G)$ of a group $\mathbf G$ is the simple digraph with vertex set $G$ in which $x\rightarrow y$ if $y$ is a power of $x$, the power graph is the underlying simple graph, and the enhanced power…
Let $P$ be a set of $n \geq 5$ points in convex position in the plane. The path graph $G(P)$ of $P$ is an abstract graph whose vertices are non-crossing spanning paths of $P$, such that two paths are adjacent if one can be obtained from the…
We examine the evolution of the best choice algorithm and the probability of its success from a directed path to the linear order of the same cardinality through $k$th powers of a directed path, $1 \leq k < n$. The vertices of a $k$th power…
In the k-Connected Directed Steiner Tree problem (k-DST), we are given a directed graph G=(V, E) with edge (or vertex) costs, a root vertex r, a set of q terminals T, and a connectivity requirement k>0; the goal is to find a minimum-cost…
We introduce a new Steiner-type problem for directed graphs named \textsc{$q$-Root Steiner Tree}. Here one is given a directed graph $G=(V,A)$ and two subsets of its vertices, $R$ of size $q$ and $T$, and the task is to find a minimum size…
We introduce the concept of a family of finite directed graphs (order a) which are directed graphs derived from an infinite directed graph (order a), called the a-root digraph. The a-root digraph has four fundamental properties which are;…
We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits countably many structures of a generalized…
We construct a family of root-finding algorithms which exploit the branched covering structure of a polynomial of degree $d$ with a path-lifting algorithm for finding individual roots. In particular, the family includes an algorithm that…