相关论文: The Toucher-Isolator Game on Trees

Let $G$ be a graph of order $n$. A classical upper bound for the domination number of a graph $G$ having no isolated vertices is $\lfloor\frac{n}{2}\rfloor$. However, for several families of graphs, we have $\gamma(G) \le…

We consider (n+1) runners with given constant unique integer speeds running along the circumference of a circle whose circumferential length is one, and all runners starting from the same point. We define and give lower bounds to a first…

For an $n$-vertex graph $G$, let $z(G;k)$ denote the number of zero forcing sets of size $k$. A conjecture of Boyer et al. asserts that the path $P_n$ maximizes these numbers coefficientwise among all $n$-vertex graphs; equivalently, the…

We study the m-eternal domination problem from the perspective of the attacker. For many graph classes, the minimum required number of guards to defend eternally is known. By definition, if the defender has less than the required number of…

We study the $(p,q)$-Maker Breaker Crossing game introduced by Day and Falgas Ravry in 'Maker-Breaker percolation games I: crossing grids'. The game described in their paper involves two players Maker and Breaker who take turns claiming p…

The reachability problem asks to decide if there exists a path from one vertex to another in a digraph. In a grid digraph, the vertices are the points of a two-dimensional square grid, and an edge can occur between a vertex and its…

Codes over trees were introduced recently to bridge graph theory and coding theory with diverse applications in computer science and beyond. A central challenge lies in determining the maximum number of labelled trees over $n$ nodes with…

Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in $\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a graph $G$, denoted by $\iota(G,\mathcal{F})$, is the size of a smallest set $D$ of vertices of…

The localization game is a pursuit-evasion game analogous to Cops and Robbers, where the robber is invisible and the cops send distance probes in an attempt to identify the location of the robber. We present a novel graph parameter called…

Given a 2-edge connected, unweighted, and undirected graph $G$ with $n$ vertices and $m$ edges, a $\sigma$-tree spanner is a spanning tree $T$ of $G$ in which the ratio between the distance in $T$ of any pair of vertices and the…

For a graph $G$, and two distinct vertices $u$ and $v$ of $G$, let $n_G(u,v)$ be the number of vertices of $G$ that are closer in $G$ to $u$ than to $v$. Miklavi\v{c} and \v{S}parl (arXiv:2011.01635v1) define the distance-unbalancedness of…

The equator of a graph is the length of a longest isometric cycle. We bound the order $n$ of a graph from below by its equator $q$, girth $g$ and minimum degree $\delta$ - and show that this bound is sharp when there exists a Moore graph…

Cops and Robbers is a game played on a graph where a set of cops attempt to capture a single robber. The game proceeds in rounds, where each round first consists of the cops' turn, followed by the robber's turn. In the cops' turn, every cop…

We study the biased $(2:b)$ Walker--Breaker games, played on the edge set of the complete graph on $n$ vertices, $K_n$. These games are a variant of the Maker--Breaker games with the restriction that Walker (playing the role of Maker) has…

We study biased Maker-Breaker games on a graph system $\{G_1,\ldots,G_s\}$, in which Maker's goal is to claim certain rainbow structures, i.e., specified subgraphs consisting of at most one edge from each graph $G_i$. We consider the…

A graph $G$ is universal for a class of graphs $\mathcal{C}$, if, up to isomorphism, $G$ contains every graph in $\mathcal{C}$ as a subgraph. In 1978, Chung and Graham asked for the minimal number $s(n)$ of edges in a graph with $n$…

Minimum spanning trees are important tools in the analysis and design of networks. Many practical applications require their computation, ranging from biology and linguistics to economy and telecommunications. The set of cycles of a network…

We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds…

Conway Checkers is a game played with a checker placed in each square of the lower half of an infinite checkerboard. Pieces move by jumping over an adjacent checker, removing the checker jumped over. Conway showed that it is not possible to…

Mader [J. Combin. Theory Ser. B 40 (1986) 152-158] proved that every $k$-edge-connected graph $G$ with minimum degree at least $k+1$ contains a vertex $u$ such that $G-\{u\}$ is still $k$-edge-connected. In this paper, we prove that every…