Related papers: New $\mathcal{F}$-Saturation Games on Directed Gra…

We study F-saturation games, first introduced by F\"uredi, Reimer and Seress in 1991, and named as such by West. The main question is to determine the length of the game whilst avoiding various classes of graph, playing on a large complete…

Given a family of graphs $\mathcal{F}$, we consider the $\mathcal{F}$-saturation game. In this game two players alternate adding edges to an initially empty graph on $n$ vertices, with the only constraint being that neither player can add…

Given a family of graphs $\mathcal{F}$, we define the $\mathcal{F}$-saturation game as follows. Two players alternate adding edges to an initially empty graph on $n$ vertices, with the only constraint being that neither player can add an…

Given a family ${\mathcal F}$ and a host graph $H$, a graph $G\subseteq H$ is ${\mathcal F}$-saturated relative to $H$ if no subgraph of $G$ lies in ${\mathcal F}$ but adding any edge from $E(H)-E(G)$ to $G$ creates such a subgraph. In the…

A graph $G = (V,E)$ is said to be saturated with respect to a monotone increasing graph property ${\mathcal P}$, if $G \notin {\mathcal P}$ but $G \cup \{e\} \in {\mathcal P}$ for every $e \in \binom{V}{2} \setminus E$. The saturation game…

We consider the following two-player game: Maxi and Mini start with the empty graph on $n$ vertices and take turns, always adding one additional edge to the graph such that the chromatic number is at most $k$, where $k \in \mathbb{N}$ is a…

This paper considers a class of two-player zero-sum games on directed graphs whose vertices are equipped with random payoffs of bounded support known by both players. Starting from a fixed vertex, players take turns to move a token along…

A graph $G$ is $F$-saturated if $G$ is $F$-free but for any edge $e$ in the complement of $G$ the graph $G + e$ contains $F$. Gerbner et al. (Discrete Math., 345 (2022), 112921) initiated the study of $rsat(n,F)$, the minimum number of…

Iterated regret minimization has been introduced recently by J.Y. Halpern and R. Pass in classical strategic games. For many games of interest, this new solution concept provides solutions that are judged more reasonable than solutions…

For a fixed finite set of finite tournaments ${\mathcal F}$, the ${\mathcal F}$-free orientation problem asks whether a given finite undirected graph $G$ has an $\mathcal F$-free orientation, i.e., whether the edges of $G$ can be oriented…

The saturation number of a graph $F$, written $\textup{sat}(n,F)$, is the minimum number of edges in an $n$-vertex $F$-saturated graph. One of the earliest results on saturation numbers is due to Erd\H{o}s, Hajnal, and Moon who determined…

Let $\mathcal{F}$ be a family of $r$-graphs. An $r$-graph $G$ is called $\mathcal{F}$-saturated if it does not contain any members of $\mathcal{F}$ but adding any edge creates a copy of some $r$-graph in $\mathcal{F}$. The saturation number…

In 1992, Bitar and Goles introduced the parallel chip-firing game on undirected graphs. Two years later, Prisner extended the game to directed graphs. While the properties of parallel chip-firing games on undirected graphs have been…

In 1964, Erd\H{o}s, Hajnal and Moon introduced a saturation version of Tur\'an's classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that…

Given positive integer $n$ and graph $F$, the saturation number $\mathrm{sat}(n, F)$ is the minimum number of edges in an edge-maximal $F$-free graph on $n$ vertices. In this paper, we determine asymptotic behavior of $\mathrm{sat}(n, F)$…

Motivated by the success of domination games and by a variation of the coloring game called the indicated coloring game, we introduce a version of domination games called the indicated domination game. It is played on an arbitrary graph $G$…

Saturation problems for forbidden graphs have been a popular area of research for many decades, and recently Brualdi and Cao initiated the study of a saturation problem for 0-1 matrices. We say that 0-1 matrix $A$ is saturating for the…

For an edge-ordered graph $G$, we say that an $n$-vertex edge-ordered graph $H$ is $G$-saturated if it is $G$-free and adding any new edge with any new label to $H$ introduces a copy of $G$. The saturation function describes the minimum…

Treewidth on undirected graphs is known to have many algorithmic applications. When considering directed width-measures there are much less results on their deployment for algorithmic results. In 2022 the first author, Rabinovich and…

The Explorer-Director game, first introduced by Nedev and Muthukrishnan, can be described as a game where two players -- Explorer and Director -- determine the movement of a token on the vertices of a graph. At each time step, the Explorer…