Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1512.02916

Progress Towards the Total Domination Game $\frac{3}{4}$-Conjecture

Michael A. Henning·Douglas F. Rall
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453--1462], where the players Dominator and Staller alternately select vertices of $G$. Each vertex chosen must strictly increase the number of vertices totally dominated, where a vertex totally dominates another vertex if they are neighbors. This process eventually produces a total dominating set $S$ of $G$ in which every vertex is totally dominated by a vertex in $S$. Dominator wishes to minimize the number of vertices chosen, while Staller wishes to maximize it. The game total domination number, $γ_{\rm tg}(G)$, of $G$ is the number of vertices chosen when Dominator starts the game and both players play optimally. Henning, Klavžar and Rall [Combinatorica, to appear] posted the $\frac{3}{4}$-Game Total Domination Conjecture that states that if $G$ is a graph on $n$ vertices in which every component contains at least three vertices, then $γ_{\rm tg}(G) \le \frac{3}{4}n$. In this paper, we prove this conjecture over the class of graphs $G$ that satisfy both the condition that the degree sum of adjacent vertices in $G$ is at least $4$ and the condition that no two vertices of degree $1$ are at distance $4$ apart in $G$. In particular, we prove that by adopting a greedy strategy, Dominator can complete the total domination game played in a graph with minimum degree at least $2$ in at most $3n/4$ moves.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michael A. Henning, Douglas F. Rall. 2015-12-09. Progress Towards the Total Domination Game $\frac{3}{4}$-Conjecture. https://arxiv.org/abs/1512.02916

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Matching Complexes of Outerplanar Graphs

An outerplanar graph is a planar graph that has a planar drawing with all vertices on the unbounded face. The matching complex of a graph is the simplicial complex whose faces are subsets of disjoint edges of the graph. In this paper we prove that the matching complexes of outerplanar graphs are contractible or homotopy equivalent to a wedge of spheres. This extends known results about trees and polygonal line tilings.

math.CO↗
arXiv2026

Awesome graph parameters

For a graph $G$, we denote by $α(G)$ the size of a maximum independent set and by $ω(G)$ the size of a maximum clique in $G$. Our paper lies on the edge of two lines of research, related to $α$ and $ω$, respectively. One of them studies $α$-variants of graph parameters, such as $α$-treewidth or $α$-degeneracy. The second line deals with graph classes where some parameters are bounded by a function of $ω(G)$. A famous example of this type is the family of $χ$-bounded classes, where the chromatic number $χ(G)$ is bounded by a function of $ω(G)$. A Ramsey-type argument implies that if the $α$-variant of a graph parameter $ρ$ is bounded by a constant in a hereditary class $\mathcal{G}$, then $ρ$ is bounded by a function of $ω$ in $\mathcal{G}$. If the reverse implication also holds, we say that $ρ$ is awesome. Otherwise, we say that $ρ$ is awful. In the present paper, we identify a number of awesome and awful graph parameters, derive some algorithmic applications of awesomeness, and propose a number of open problems related to these notions.

math.CO↗
arXiv2026

Perfect matchings and $A_α$-spectral radius in 1-binding graphs

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For $α\in[0,1)$, we use $A_α(G)$ and $ρ_α(G)$ to denote the $A_α$-matrix and the $A_α$-spectral radius of $G$, respectively. The binding number $\mbox{bind}(G)$ of $G$ is defined by $\mbox{bind}(G)=\min\left\{\frac{|N_G(X)|}{|X|}:\emptyset\neq X\subseteq V(G),N_G(X)\neq V(G)\right\}$. If $\mbox{bind}(G)\geq1$, then $G$ is called 1-binding. A perfect matching in $G$ is a set of nonadjacent edges covering every vertex of $G$. Tutte proved that a graph $G$ of even order has a perfect matching if and only if $o(G-S)\leq|S|$ holds for every $S\subseteq V(G)$ [W. Tutte, The factorization of linear graphs, J. Lond. Math. Soc. 22 (1947) 107--111]. In this paper, we use Tutte's result to prove that a connected 1-binding graph $G$ of even order $n$ with $n\geq n(α)$ has a perfect matching unless $G=K_1\vee(K_{n-5}\cup K_3\cup K_1)$ if $ρ_α(G)\geqρ_α(K_1\vee(K_{n-5}\cup K_3\cup K_1))$, where $n(α)$ is defined as follows: $n(α)=\max\{18,\frac{2+8α}{1-2α}\}$ if $α\in[0,\frac{1}{2})$, and $n(α)=18$ if $α=\frac{1}{2}$.

math.CO↗