相关论文: Transversals of Longest Paths and Cycles

In 2015, Dankelmann and Bau proved that for every bridgeless graph $G$ of order $n$ and minimum degree $\delta$ there is an orientation of diameter at most $11\frac{n}{\delta+1}+9$. In 2016, Surmacs reduced this bound to…

A path $P$ in an edge-colored graph $G$ is called a proper path if no two adjacent edges of $P$ are colored the same, and $G$ is proper connected if every two vertices of $G$ are connected by a proper path in $G$. The proper connection…

In this paper, we study the outerplanarity of planar graphs, i.e., the number of times that we must (in a planar embedding that we can initially freely choose) remove the outerface vertices until the graph is empty. It is well-known that…

Given a connected graph $G$, the metric (resp. edge metric) dimension of $G$ is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of $G$ by means of distance…

A simple topological graph T = (V(T), E(T)) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H…

A $t$-bar visibility representation of a graph assigns each vertex up to $t$ horizontal bars in the plane so that two vertices are adjacent if and only if some bar for one vertex can see some bar for the other via an unobstructed vertical…

The leafage l(G) of a chordal graph G is the minimum number of leaves of a tree in which G has an intersection representation by subtrees. We obtain upper and lower bounds on l(G) and compute it on special classes. The maximum of l(G) on…

Let G be a graph of order n and let q(G) be that largest eigenvalue of the signless Laplacian of G. In this note it is shown that if k>1 and q(G)>=n+2k-2, then G contains cycles of length l whenever 2<l<2k+3. This bound is asymptotically…

Given a simple graph $G$, its line graph, denoted by $L(G)$, is obtained by representing each edge of $G$ as a vertex, with two vertices in $L(G)$ adjacent whenever the corresponding edges in $G$ share a common endpoint. By applying the…

A minimum feedback arc set of a directed graph $G$ is a smallest set of arcs whose removal makes $G$ acyclic. Its cardinality is denoted by $\beta(G)$. We show that an Eulerian digraph with $n$ vertices and $m$ arcs has $\beta(G) \ge…

The outer multiset dimension ${\rm dim}_{\rm ms}(G)$ of a graph $G$ is the cardinality of a smallest set of vertices that uniquely recognize all the vertices outside this set by using multisets of distances to the set. It is proved that…

The segment number of a planar graph $G$ is the smallest number of line segments needed for a planar straight-line drawing of $G$. Dujmovi\'c, Eppstein, Suderman, and Wood [CGTA'07] introduced this measure for the visual complexity of…

In this paper, we obtain lower bounds for the domination numbers of connected graphs with girth at least $7$. We show that the domination number of a connected graph with girth at least $7$ is either $1$ or at least…

We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most…

A separating path system for a graph $G$ is a collection $\mathcal{P}$ of paths in $G$ such that for every two edges $e$ and $f$ in $G$, there is a path in $\mathcal{P}$ that contains $e$ but not $f$. We show that every $n$-vertex graph has…

In this paper, we relate the seemingly unrelated concepts of treewidth and boxicity. Our main result is that, for any graph G, boxicity(G) <= treewidth(G) + 2. We also show that this upper bound is (almost) tight. Our result leads to…

The following theorem is proved: For all $k$-connected graphs $G$ and $H$ each with at least $n$ vertices, the treewidth of the cartesian product of $G$ and $H$ is at least $k(n -2k+2)-1$. For $n\gg k$ this lower bound is asymptotically…

A covering path for a finite set $P$ of points in the plane is a polygonal path such that every point of $P$ lies on a segment of the path. The vertices of the path need not be at points of $P$. A covering path is plane if its segments do…

The relation between the Wiener index $W(G)$ and the eccentricity $\varepsilon(G)$ of a graph $G$ is studied. Lower and upper bounds on $W(G)$ in terms of $\varepsilon(G)$ are proved and extremal graphs characterized. A Nordhaus-Gaddum type…

The treewidth of a graph is an important invariant in structural and algorithmic graph theory. This paper studies the treewidth of line graphs. We show that determining the treewidth of the line graph of a graph $G$ is equivalent to…