Related papers: Path eccentricity of graphs
Motivated by a relaxed notion of the celebrated Hamiltonian cycle, this paper investigates its variant, parity Hamiltonian cycle (PHC): A PHC of a graph is a closed walk which visits every vertex an odd number of times, where we remark that…
The transmission of a vertex $v$ of a graph $G$ is the sum of distances from $v$ to all the other vertices in $G$. The Wiener complexity of $G$ is the number of different transmissions of its vertices. Similarly, the eccentric complexity of…
The complexity of the maximum common connected subgraph problem in partial $k$-trees is still not fully understood. Polynomial-time solutions are known for degree-bounded outerplanar graphs, a subclass of the partial $2$-trees. On the other…
Matrix-based centrality measures have enjoyed significant popularity in network analysis, in no small part due to our ability to rigorously analyze their behavior as parameters vary. Recent work has considered the relationship between…
In the classical Node-Disjoint Paths (NDP) problem, the input consists of an undirected $n$-vertex graph $G$, and a collection $\mathcal{M}=\{(s_1,t_1),\ldots,(s_k,t_k)\}$ of pairs of its vertices, called source-destination, or demand,…
A mobile agent, starting from a node $s$ of a simple undirected connected graph $G=(V,E)$, has to explore all nodes and edges of $G$ using the minimum number of edge traversals. To do so, the agent uses a deterministic algorithm that allows…
Among the most important graph parameters is the Diameter, the largest distance between any two vertices. There are no known very efficient algorithms for computing the Diameter exactly. Thus, much research has been devoted to how fast this…
A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored the same. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…
We consider the problem of partitioning the edges of a graph into as few paths as possible. This is a~subject of the classic conjecture of Gallai and a recurring topic in combinatorics. Regarding the complexity of partitioning a graph…
We introduce and study "path odd-covers", a weakening of Gallai's path decomposition problem and a strengthening of the linear arboricity problem. The "path odd-cover number" $p_2(G)$ of a graph $G$ is the minimum cardinality of a…
The proximity $\pi = \pi (G)$ of a connected graph $G$ is the minimum, over all vertices, of the average distance from a vertex to all others. Similarly, the maximum is called the remoteness and denoted by $\rho = \rho (G)$. The concepts of…
We study the asymptotic behavior of the number of paths of length $N$ on several classes of infinite graphs with a single special vertex. This vertex can work as an entropic trap for the path, i.e. under certain conditions the dominant part…
A set of vertices $S$ \emph{resolves} a connected graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The \emph{metric dimension} of $G$ is the minimum cardinality of a resolving set of $G$.…
Graphs are fundamental objects that find widespread applications across computer science and beyond. Graph Theory has yielded deep insights about structural properties of various families of graphs, which are leveraged in the design and…
The purpose of this work is to introduce and characterize the Bounded Acceleration Shortest Path (BASP) problem, a generalization of the Shortest Path (SP) problem. This problem is associated to a graph: the nodes represent positions of a…
We study a large family of graph covering problems, whose definitions rely on distances, for graphs of bounded cyclomatic number (that is, the minimum number of edges that need to be removed from the graph to destroy all cycles). These…
In the past decades, many scholars concerned which edge-extremal problems have spectral analogues? Recently, Wang, Kang and Xue showed an interesting result on $F$-free graphs [J. Combin. Theory Ser. B 159 (2023) 20--41]. In this paper, we…
Computing a (short) path between two vertices is one of the most fundamental primitives in graph algorithmics. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set is fixed but the edge set changes…
The study of vertex centrality measures is a key aspect of network analysis. Naturally, such centrality measures have been generalized to groups of vertices; for popular measures it was shown that the problem of finding the most central…
This paper develops a structural theory of unique shortest paths in real-weighted graphs. Our main goal is to characterize exactly which sets of node sequences, which we call path systems, can be realized as unique shortest paths in a graph…