The graph grabbing game on blow-ups of trees and cycles

A vertex $v$ of a connected graph $G$ is a cut vertex if $G-v$ is disconnected. A graph $G$ is even (resp. odd) if the number of vertices of $G$ is even (resp. odd). A weighted graph $G$ is a graph $G$ with a weighted function $w:V(G)\rightarrow\mathbb{R}^{+}\cup\{0\}$.

The graph grabbing game is played on a non-negatively weighted connected graph by two players: Alice and Bob who alternately claim a non-cut vertex from the remaining graph and collect the weight on the vertex, where Alice plays first. The aim of each player is to maximize the weights on their respective claimed vertices at the end of the game when all vertices have been claimed. Alice wins the game if she gains at least half of the total weight of the graph.

The first version of the graph grabbing game appeared in the first problem in Winkler’s puzzle book (2003) [Winkler], where he gave a winning strategy for Alice on every weighted even path and he observed that there is a weighted odd path on which Alice cannot win. In 2009, Rosenfeld [Rosenfeld:gold] proposed the game for trees and call it the gold grabbing game. In 2011, Micek and Walczak [Micek:subdividedstar] generalized the game to general graphs and call it the graph grabbing game. They showed that Alice can secure at least a quarter of the weight of every weighted even tree and they conjectured that Alice can in fact secure at least half of the weight of every weighted even tree. Later in 2012, Seacrest and Seacrest [Seacrest:tree] solved this conjecture by considering a vertex-rooted version of the game and they posed the following conjecture.

In 2018, Egawa, Enomoto and Matsumoto [Egawa:Kmntree] gave a supporting evidence for this conjecture. They generalized the proof of Seacrest and Seacrest by considering a set-rooted version of the game to prove that Alice wins the game on every weighted even $K_{m,n}$-tree, namely a bipartite graph obtained from a complete bipartite graph $K_{m,n}$ on $[m+n]$ and trees $T_{1},\dots,T_{m+n}$ by identifying vertex $i$ of $K_{m,n}$ with exactly one vertex of $T_{i}$ for each $i\in[m+n]$.

For a graph $G$ with vertices $v_{1},\dots,v_{k}$ and non-empty sets $V_{1},\dots,V_{k}$, a blow-up B($G$) of $G$ is a graph obtained from $G$ by replacing $v_{1},\dots,v_{k}$ with $V_{1},\dots,V_{k}$, respectively where, for each $i,j\in[k]$, vertices $x\in V_{i}$ and $y\in V_{j}$ are adjacent in B($G$) if and only if $v_{i}$ and $v_{j}$ are adjacent in $G$. For a graph $G$ on $[k]$ and trees $T_{1},\dots,T_{k}$, a $G$-tree is a graph obtained from $G$ by identifying vertex $i$ of $G$ with exactly one vertex of $T_{i}$ for each $i\in[k]$. For a tree $T$, we note that a B$(T)$-tree and B($C_{2n}$) are connected bipartite graphs, and a B$(T)$-tree is a $K_{m,n}$-tree when $T$ is the path on two vertices.

In this paper, we partially confirm Conjecture 1 as follows.

For a graph $G$ and a set $S\subseteq V(G)$, let $N_{G}(S)$ denote the neighborhood of $S$, i.e. the set of vertices having a neighbor in $S$. The proof is based on the method of Egawa, Enomoto and Matsumoto, where their main lemmas dealt with the score of the game on a $K_{m,n}$-tree rooted at a partite class. We generalize their method by considering instead the scores of the game on a $H$-tree rooted at $V_{i}$ and the game on $H$-tree rooted at $N_{H}(V_{i})$, where $H$ is a blow-up of a tree.

The rest of this paper is organized as follows. In Section 2, we recall some observations and a lemma on $K_{m,n}$-trees given by Egawa, Enomoto and Matsumoto. Section 3 is devoted to proving Theorem 2 and then applying it to prove Corollary 3. In Section LABEL:section-conclusion, we give some concluding remarks.

In this section, we prepare some observations and a lemma on $K_{m,n}$-trees which will be useful for the proof of Theorem 2.

We first give definitions of a rooted version of the graph grabbing game and some related terms introduced by Egawa, Enomoto and Matsumoto. For a weighted graph $G$, a root set $S$ of $G$ is a set of vertices intersecting every component of $G$ and the game on $G$ rooted at $S$ is a graph grabbing game, where each player needs not claim a non-cut vertex, but instead they claim a vertex $v$ such that every component of $G-v$ contains at least one vertex in $S$. Therefore, a move $v$ in the game on $G$ is feasible if $G-v$ is connected, and a move $v$ in the game on $G$ rooted at $S$ is feasible if every component of $G-v$ contains at least one vertex in $S$. A move $v$ in the game on $G$ (rooted at $S$) is optimal if there is an optimal strategy in the game on $G$ (rooted at $S$) having $v$ as the first move. The first (resp. second) player is called Player $1$ (resp. Player $2$). The last (resp. second from last) player is called Player $-1$ (resp. Player $-2$). For $k\in\{1,2,-1,-2\}$, assuming that both players play optimally, let $N(G,k)$ denote the score of Player $k$ in the game on $G$ and let $R(G,S,k)$ denote the score of Player $k$ in the game on $G$ rooted at $S$ and we write $R(G,v,k)$ for $R(G,\{v\},k)$. For a set $S$ and an element $x$, we write $S-x$ for $S\setminus\{x\}$.

Egawa, Enomoto and Matsumoto observed some relationships between the scores of both players in the normal version and the rooted version of the game. Note that the equation/inequality in the brackets in each observation is an equivalent form of the first one due to the fact that, assuming that both players play optimally, the sum of their scores equals to the total weight of the graph.

The next lemma is a part of their main results which will help us in the proof.

In this section, we start by proving Lemma 8 which will be used repeatedly in the proof of our main lemmas, namely, Lemmas LABEL:lem1 and LABEL:lem2. We then prove Theorem 2 by applying the main lemmas and deduce Corollary 3 from Theorem 2.

The following lemma shows the relationship between the scores of both players in the game on an even graph rooted at two different sets of some structure.