The graph grabbing game on blow-ups of trees and cycles
aa r X i v : . [ m a t h . C O ] J u l The graph grabbing game on blow-ups of trees and cycles
Sopon Boriboon ∗ Teeradej Kittipassorn † July 24, 2020
Abstract
The graph grabbing game is played on a non-negatively weighted connected graph byAlice and Bob who alternately claim a non-cut vertex from the remaining graph, where Aliceplays first, to maximize the weights on their respective claimed vertices at the end of thegame when all vertices have been claimed. Seacrest and Seacrest conjectured that Alice cansecure at least half of the weight of every weighted connected bipartite even graph. Later,Egawa, Enomoto and Matsumoto partially confirmed this conjecture by showing that Alicewins the game on a class of weighted connected bipartite even graphs called K m,n -trees. Weextend the result on this class to include a number of graphs, e.g. even blow-ups of treesand 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 ) → R + ∪ { } .The graph grabbing game is played on a non-negatively weighted connected graph by twoplayers: Alice and Bob who alternately claim a non-cut vertex from the remaining graph andcollect the weight on the vertex, where Alice plays first. The aim of each player is to maximizethe weights on their respective claimed vertices at the end of the game when all vertices havebeen 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 puzzlebook (2003) [12], where he gave a winning strategy for Alice on every weighted even path and heobserved that there is a weighted odd path on which Alice cannot win. In 2009, Rosenfeld [10]proposed the game for trees and call it the gold grabbing game . In 2011, Micek and Walczak [8] ∗ Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok10330, Thailand; [email protected] . † Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok10330, Thailand; [email protected] . Conjecture 1 ([11]) . Alice wins the game on every weighted connected bipartite even graph.
In 2018, Egawa, Enomoto and Matsumoto [3] gave a supporting evidence for this conjecture.They generalized the proof of Seacrest and Seacrest by considering a set-rooted version of thegame to prove that Alice wins the game on every weighted even K m,n -tree , namely a bipartitegraph obtained from a complete bipartite graph K m,n on [ m + n ] and trees T , . . . , T m + n byidentifying vertex i of K m,n with exactly one vertex of T i for each i ∈ [ m + n ] .For a graph G with vertices v , . . . , v k and non-empty sets V , . . . , V k , a blow-up B( G ) of G is a graph obtained from G by replacing v , . . . , v k with V , . . . , V k , respectively where, for each i, j ∈ [ k ] , vertices x ∈ V i and y ∈ 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 , . . . , T k , a G -tree is a graph obtained from G by identifyingvertex i of G with exactly one vertex of T i for each i ∈ [ k ] . For a tree T , we note that a B ( T ) -treeand B( C n ) are connected bipartite graphs, and a B ( T ) -tree is a K m,n -tree when T is the pathon two vertices. v v v v v v v T V V V V V V V B ( T ) B ( T ) -treeFigure 1: Examples of a tree T , a blow-up B ( T ) and a B ( T ) -tree.In this paper, we partially confirm Conjecture 1 as follows. Theorem 2.
Alice wins the game on every weighted even B ( T ) -tree, where T is a tree. Corollary 3.
Alice wins the game on every weighted even B ( C n ) . For a graph G and a set S ⊆ V ( G ) , let N G ( S ) denote the neighborhood of S , i.e. the setof vertices having a neighbor in S . The proof is based on the method of Egawa, Enomoto andMatsumoto, where their main lemmas dealt with the score of the game on a K m,n -tree rooted at2 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 alemma on K m,n -trees given by Egawa, Enomoto and Matsumoto. Section 3 is devoted to provingTheorem 2 and then applying it to prove Corollary 3. In Section 4, we give some concludingremarks. In this section, we prepare some observations and a lemma on K m,n -trees which will be usefulfor the proof of Theorem 2.We first give definitions of a rooted version of the graph grabbing game and some relatedterms 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 graphgrabbing game, where each player needs not claim a non-cut vertex, but instead they claim avertex 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 (resp. Player ). The last(resp. second from last) player is called Player − (resp. Player − ). For k ∈ { , , − , − } ,assuming that both players play optimally, let N ( G, k ) denote the score of Player k in the gameon 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 \ { x } .Egawa, Enomoto and Matsumoto observed some relationships between the scores of both play-ers in the normal version and the rooted version of the game. Note that the equation/inequalityin 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 ofthe graph. Observation 4 ([3]) . If x is a feasible move in the game on G , then N ( G, ≤ N ( G − x,
1) ( ⇔ N ( G, ≥ N ( G − x,
2) + w ( x )) .If x is an optimal move in the game on G , then N ( G,
2) = N ( G − x,
1) ( ⇔ N ( G,
1) = N ( G − x,
2) + w ( x )) . Observation 5 ([3]) . Let S be a root set of G . If x is a feasible move in the game on G rootedat S , then ( G, S, ≤ R ( G − x, S − x,
1) ( ⇔ R ( G, S, ≥ R ( G − x, S − x,
2) + w ( x )) .If x is an optimal move in the game on G rooted at S , then R ( G, S,
2) = R ( G − x, S − x,
1) ( ⇔ R ( G, S,
1) = R ( G − x, S − x,
2) + w ( x )) . Observation 6 ([3]) . If v is a root of G , then R ( G, v, −
2) = R ( G − v, N G ( v ) , −
1) ( ⇔ R ( G, v, −
1) = R ( G − v, N G ( v ) , −
2) + w ( v )) . The next lemma is a part of their main results which will help us in the proof.
Lemma 7 ([3]) . Let G be a K m,n -tree with partite classes X, Y of size m, n ≥ , respectively.Then R ( G, Y, − ≤ N ( G, −
2) ( ⇔ R ( G, Y, − ≥ N ( G, − . In this section, we start by proving Lemma 8 which will be used repeatedly in the proof ofour main lemmas, namely, Lemmas 9 and 10. We then prove Theorem 2 by applying the mainlemmas and deduce Corollary 3 from Theorem 2.The following lemma shows the relationship between the scores of both players in the gameon an even graph rooted at two different sets of some structure.
Lemma 8.
Let G and G be subgraphs of an even graph G such that V ( G ) , V ( G ) partition V ( G ) . If U = V ( G ) ∩ N G ( V ( G )) and U = V ( G ) ∩ N G ( V ( G )) are root sets of G and G ,respectively, and every vertex in U is joined to every vertex in U , then8.1 R ( G, U , ≥ R ( G , U , −
2) + R ( G , U , − .8.2 R ( G, U , ≥ R ( G, U , U U G G Figure 2: The graph G in Lemma 8.4 roof. First, we shall prove Lemma 8.1 by considering a strategy for Alice who plays first in thegame on G rooted at U . She plays optimally as Player − in the game on G rooted at U and plays optimally as Player − in the game on G rooted at U . Since | V ( G ) | + | V ( G ) | iseven, she plays as Player in one game and as Player in the other. Now, we check that Alice’smoves are feasible in the game on G rooted at U , and Bob’s moves are feasible in the gameon G rooted at U and the game on G rooted at U . Indeed, after each move of Alice, everyremaining component of G and G contains a vertex in U and U , respectively. Together withthe fact that every vertex in U is joined to the remaining subset of U , we can conclude thatevery remaining component of G contains a vertex in U . That is, her moves are feasible in thegame on G rooted at U . On the other hand, after each move of Bob, every remaining componentof G contains a vertex of U . Since the edges between G and G have endpoints only in U and U , every remaining component of G or G contains a vertex in U or U , respectively. That is,his moves are feasible in the game on G rooted at U and the game on G rooted at U . Hence R ( G, U , ≥ R ( G , U , −
2) + R ( G , U , − , which completes the proof of Lemma 8.1. By symmetry, we have R ( G, U , ≥ R ( G , U , −
1) + R ( G , U , − , which is equivalent to R ( G, U , ≤ R ( G , U , −
2) + R ( G , U , − , by considering the total weight of G, G and G . Together with Lemma 8.1, we have R ( G, U , ≤ R ( G , U , −
2) + R ( G , U , − ≤ R ( G, U , , which completes the proof of Lemma 8.2.We are now ready to prove the main lemmas which generalize the results on K m,n -trees toB ( T ) -trees relating the scores of both players in the normal version and the rooted version of thegame. Lemma 9.
Let H be a blow-up graph of a tree with sets of vertices V , . . . , V k and let G be a H -tree.9.1 For a vertex v ∈ V ( G ) , R ( G, v, − ≤ N ( G, −
2) ( ⇔ R ( G, v, − ≥ N ( G, − .9.2 For each i ∈ [ k ] , R ( G, V i , − ≤ N ( G, −
2) ( ⇔ R ( G, V i , − ≥ N ( G, − .9.3 For each i ∈ [ k ] , R ( G, N H ( V i ) , − ≤ N ( G, −
2) ( ⇔ R ( G, N H ( V i ) , − ≥ N ( G, − . Lemma 10.
Let H be a blow-up graph of a tree with sets of vertices V , . . . , V k and let G be aneven H -tree. v ∈ V ( G ) , R ( G, v, ≥ N ( G,
2) ( ⇔ R ( G, v, ≤ N ( G, .10.2 For each i ∈ [ k ] , R ( G, V i , ≥ N ( G,
2) ( ⇔ R ( G, V i , ≤ N ( G, .10.3 For each i ∈ [ k ] , R ( G, N H ( V i ) , ≥ N ( G,
2) ( ⇔ R ( G, N H ( V i ) , ≤ N ( G, . We prove Lemmas 9 and 10 simultaneously by induction on the number n of vertices of G .It is easy to check that Lemmas 9 and 10 hold for n ≤ . Now, we let n ≥ and suppose thatLemmas 9 and 10 hold for | V ( G ) | < n . We remark that the following fact will be used throughoutthe proofs: Let G be a H -tree, where H is a blow-up of a tree and let v be a vertex in G . Then G − v is a H ′ -tree, where H ′ is a blow-up of some tree if and only if G − v is connected. Proof of Lemma 9.1.
Let v ∈ V ( G ) . Case 1. G is even.Let a be an optimal move in the game on G rooted at v . Therefore, a = v and a is feasiblein the game on G . So G − a is connected. Then R ( G, v, − R ( G − a, v, −
1) (
Observation 5 ) ≥ N ( G − a, − Lemma 9.1 by induction ) ≥ N ( G, −
1) (
Observation 4 ) . Case 2. G is odd.Let b be an optimal move in the game on G . So G − b is connected. Case 2.1. b = v .Therefore, b is a feasible move in the game on G rooted at v . Then R ( G, v, − ≤ R ( G − b, v, −
2) (
Observation 5 ) ≤ N ( G − b, − Lemma 9.1 by induction )= N ( G, −
2) (
Observation 4 ) . Case 2.2. b = v and v is a leaf.Let u be the unique neighbor of v . Then R ( G, v, −
2) = R ( G − v, u, − Observation 6 ) ≤ N ( G − v,
1) (
Lemma 10.1 by induction )= N ( G, −
2) (
Observation 4 and b = v ) . Case 2.3. b = v and v is not a leaf. 6herefore, v ∈ V i for some i ∈ [ k ] and N G ( v ) = N H ( V i ) . Then R ( G, v, −
2) = R ( G − v, N G ( v ) = N H ( V i ) , − Observation 6 ) ≤ N ( G − v,
1) (
Lemma 10.3 by induction )= N ( G, −
2) (
Observation 4 and b = v ) . Proof of Lemma 9.2.
Let i ∈ [ k ] . If | V i | = 1 , then we are done by Lemma 9.1. Now, supposethat | V i | ≥ . Case 1. G is odd.Let b be an optimal move in the game on G . So G − b is connected. Since | V i | ≥ , we have V i − b = ∅ . Therefore, b is a feasible move in the game on G rooted at V i . Then N ( G, − N ( G − b, −
2) (
Observation 4 ) ≥ R ( G − b, V i − b, − Lemma 9.2 by induction ) ≥ R ( G, V i , −
2) (
Observation 5 ) . Case 2. G is even.Let a be an optimal move in the game on G rooted at V i . Case 2.1. a is a feasible move in the game on G .So G − a is connected. Then R ( G, V i , − R ( G − a, V i − a, −
1) (
Observation 5 ) ≥ N ( G − a, − Lemma 9.2 by induction ) ≥ N ( G, −
1) (
Observation 4 ) . Case 2.2. a is not a feasible move in the game on G . aV j V i Figure 3: The graph G in Case 2.2 of Lemma 9.2.So G − a is disconnected. Since a is a feasible move in the game on G rooted at V i , we have a ∈ V j for some j ∈ [ k ] and N G ( V j ) = N H ( V j ) . Since G − a is disconnected, V j = { a } and a is not a leaf. If i = j , then every component of G − a does not contain a vertex in V i . If there7s a vertex set V l , where l / ∈ { i, j } , then either G − a is connected or there is a component of G − a which does not contain a vertex in V i . Hence V j = { a } for some j = i , N H ( V j ) = V i and N H ( V i ) = V j . Therefore, G is a K m,n -tree with partite classes V i , V j . Then, by Lemma 7, N ( G, − ≤ R ( G, V i , − . Proof of Lemma 9.3.
Let i ∈ [ k ] . If | N H ( V i ) | = 1 or N H ( V i ) = V j for some j ∈ [ k ] , then we aredone by Lemmas 9.1 or 9.2, respectively. Now, suppose that | N H ( V i ) | ≥ and V i is joined to atleast two sets in V , . . . , V k . Case 1. G is odd.Let b be an optimal move in the game on G . So G − b is connected. Since | N H ( V i ) | ≥ , wehave N H ( V i ) − b = ∅ . Then b is a feasible move in the game on G rooted at N H ( V i ) . Then N ( G, − N ( G − b, −
2) (
Observation 4 ) ≥ R ( G − b, N H ( V i ) − b, − Lemma 9.3 by induction ) ≥ R ( G, N H ( V i ) , −
2) (
Observation 5 ) . Case 2. G is even.Let a be an optimal move in the game on G rooted at N H ( V i ) . Case 2.1. a is a feasible move in the game on G .So G − a is connected. Then R ( G, N H ( V i ) , − R ( G − a, N H ( V i ) − a, −
1) (
Observation 5 ) ≥ N ( G − a, − Lemma 9.3 by induction ) ≥ N ( G, −
1) (
Observation 4 ) . Case 2.2. a is not a feasible move in the game on G . V j N H ( V i ) \ V j G = H G H aV i Figure 4: The graph G in Case 2.2 of Lemma 9.3.8o G − a is disconnected. Since a is a feasible move in the game on G rooted at N H ( V i ) , wehave a ∈ V j for some j ∈ [ k ] and N G ( V j ) = N H ( V j ) . Since G − a is disconnected, V j = { a } and a is not a leaf. Suppose that i = j . Since V i is joined to at least two sets, V i and N H ( V i ) lie inthe same component of G − a , but then the other components of G − a does not contain a vertexin N H ( V i ) , which is a contradiction. Hence V i = { a } . Let V j ⊆ N H ( V i ) and let G be the unionof components in G − a containing some vertices of V j and let G = G − a − G . By assumption, G is not empty.First, we shall show that R ( G, N H ( V i ) , − ≥ R ( G , V j , −
1) + R ( G , N H ( V i ) \ V j , − ,by considering a strategy for Bob who plays second in the game on G rooted at N H ( V i ) afterAlice grabs a . He plays optimally as Player − in the game on G rooted at V j and playsoptimally as Player − in the game on G rooted at N H ( V i ) \ V j . Since | V ( G ) | + | V ( G ) | is odd,he plays as Player in one game and as Player in the other. Now, we check that Bob’s movesare feasible in the game on G rooted at N H ( V i ) and Alice’s moves are feasible in the game on G rooted at V j and the game on G rooted at N H ( V i ) \ V j . Indeed, after each move of Bob, everyremaining component in G or G contains a vertex in V j or N H ( V i ) \ V j , respectively. Thenevery remaining component of G contains a vertex in N H ( V i ) . That is, his moves are feasible inthe game on G rooted at N H ( V i ) . On the other hand, after each move of Alice, every remainingcomponent of G contains a vertex in N H ( V i ) . Then every remaining component of G or G contains a vertex in V j or N H ( V i ) \ V j , respectively. That is, her moves are feasible in the gameon G rooted at V j and the game on G rooted at N H ( V i ) \ V j . Hence R ( G, N H ( V i ) , − ≥ R ( G , V j , −
1) + R ( G , N H ( V i ) \ V j , − . (1)Next, we let H = G and H = G − G . We observe that V j = V ( H ) ∩ N G ( V ( H )) and { a } = V ( H ) ∩ N G ( V ( H )) are root sets of H and H , respectively, and a is adjacent to allvertices in V j . Hence R ( G, V j , − ≥ R ( G , V j , −
2) + R ( G − G , a, −
1) (
Lemma 8.1 )= R ( G , V j , −
2) + R ( G , N H ( V i ) \ V j , −
2) + w ( a ) ( Observation 6 ) , which is equivalent to R ( G, V j , − ≤ R ( G , V j , −
1) + R ( G , N H ( V i ) \ V j , − , (2)by considering the total weight of G, G and G . Then N ( G, − ≤ R ( G, V j , −
1) (
Lemma 9.2 ) ≤ R ( G , V j , −
1) + R ( G , N H ( V i ) \ V j , −
1) (
Inequality 2 ) ≤ R ( G, N H ( V i ) , −
1) (
Inequality 1 ) . i N H ( V i ) G G Figure 5: The graph G in Lemma 10.3. Proof of Lemma 10.3.
For i ∈ [ k ] , let G be the union of components of G − V i containing somevertices of N H ( V i ) and let G = G − G . We observe that N H ( V i ) = V ( G ) ∩ N G ( V ( G )) and V i = V ( G ) ∩ N G ( V ( G )) are root sets of G and G , respectively, and every vertex in N H ( V i ) is joined to every vertex in V i . Then N ( G, − ≤ R ( G, V i , − Lemma 9.2 ) ≤ R ( G, N H ( V i ) ,
1) (
Lemma 8.2 ) . Proof of Lemma 10.2.
For i ∈ [ k ] , let G be the union of components of G − N H ( V i ) containingsome vertices of V i and let G = G − G . We observe that V i = V ( G ) ∩ N G ( V ( G )) and N H ( V i ) = V ( G ) ∩ N G ( V ( G )) are root sets of G and G , respectively, and every vertex in V i is joined to every vertex in N H ( V i ) . Then N ( G, − ≤ R ( G, N H ( V i ) , − Lemma 9.3 ) ≤ R ( G, V i ,
1) (
Lemma 8.2 ) . Proof of Lemma 10.1.
Let v ∈ V ( G ) . Case 1.
There is a cut edge uv incident to v . G G v u Figure 6: The graph G in Case 1 of Lemma 10.1.Let G be the component of G − uv containing v and let G = G − G . We observe that { v } = V ( G ) ∩ N G ( V ( G )) and { u } = V ( G ) ∩ N G ( V ( G )) are root sets of G and G , respectively,and v is adjacent to u . Then R ( G, v, ≥ R ( G, u, −
1) (
Lemma 8.2 ) N ( G, − Lemma 9.1 ) . Case 2.
There is no cut edge incident to v .Then v ∈ V j for some j ∈ [ k ] and N G ( v ) = N H ( V j ) . Case 2.1. | V j | ≥ .Therefore, v is a feasible move in the game on G . So G − v is connected. Then R ( G, v, −
2) = R ( G − v, N G ( v ) = N H ( V j ) , −
1) (
Observation 6 ) ≥ N ( G − v, − Lemma 9.3 by induction ) ≥ N ( G,
2) (
Observation 4 ) . Case 2.2. | V j | = 1 .Then, by Lemma 10.2, R ( G, v,
1) = R ( G, V j , ≥ N ( G, . We proceed to prove our main theorem.
Proof of Theorem 2.
Let G be an even B ( T ) -tree, where T is a tree and let v ∈ V ( G ) . Then, byLemmas 9.1 and 10.1, it follows that N ( G, − ≤ R ( G, v, − ≤ N ( G, . Therefore, Alice wins the game on G .We now deduce Corollary 3 from Theorem 2. Proof of Corollary 3.
We give a proof by induction on the number of vertices. Let G be an evenblow-up of a cycle. We note that every vertex of G is a non-cut vertex. Alice claims a maximumweighted vertex of G in her first move, say a vertex a . Let b be the vertex claimed by Bob inhis first move. Then G − { a, b } is an even blow-up of either a path or a cycle. If G − { a, b } is aneven blow-up of a path, then Alice wins the game on G − { a, b } by Theorem 2. Otherwise, Alicewins the game on G − { a, b } by the induction hypothesis. In both cases, since w ( a ) ≥ w ( b ) , Alicewins the game on G . We provide two new classes, namely B ( T ) -trees and B ( C n ) , of bipartite even graphs whichsatisfy Conjecture 1. However, this conjecture is still open. It was shown in [3] that Lemmas 9.111nd 10.1 are not true for general bipartite graphs, therefore this method cannot be directly usedto solve the full conjecture. There are several variants of the graph grabbing game, for example,the graph sharing game (see [1, 2, 5, 6, 9]), the graph grabbing game on { , } -weighted graphs(see [4]), and the convex grabbing game (see [7]), where a few problems were left open. Acknowledgment
The first author is grateful for financial support from the Science Achievement Scholarship ofThailand.
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