We study the Maker-Breaker domination game played by Dominator and Staller on the vertex set of a given graph. Dominator wins when the vertices he has claimed form a dominating set of the graph. Staller wins if she makes it impossible for Dominator to win, or equivalently, she is able to claim some vertex and all its neighbours. Maker-Breaker domination number $gamma_{MB}(G)$ ($gamma '_{MB}(G)$) of a graph $G$ is defined to be the minimum number of moves for Dominator to guarantee his winning when he plays first (second). We investigate these two invariants for the Cartesian product of any two graphs. We obtain upper bounds for the Maker-Breaker domination number of the Cartesian product of two arbitrary graphs. Also, we give upper bounds for the Maker-Breaker domination number of the Cartesian product of the complete graph with two vertices and an arbitrary graph. Most importantly, we prove that $gamma'_{MB}(P_2square P_n)=n$ for $ngeq 1$, $gamma_{MB}(P_2square P_n)$ equals $n$, $n-1$, $n-2$, for $1leq nleq 4$, $5leq nleq 12$, and $ngeq 13$, respectively. For the disjoint union of $P_2square P_n$s, we show that $gamma_{MB}'(dotcup_{i=1}^k(P_2square P_n)_i)=kcdot n$ ($ngeq 1$), and that $gamma_{MB}(dotcup_{i=1}^k(P_2square P_n)_i)$ equals $kcdot n$, $kcdot n-1$, $kcdot n-2$ for $1leq nleq 4$, $5leq nleq 12$, and $ngeq 13$, respectively.
{"title":"Maker-Breaker domination number for Cartesian products of path graphs $P_2$ and $P_n$","authors":"J. Forcan, Jiayue Qi","doi":"10.46298/dmtcs.10465","DOIUrl":"https://doi.org/10.46298/dmtcs.10465","url":null,"abstract":"We study the Maker-Breaker domination game played by Dominator and Staller on\u0000the vertex set of a given graph. Dominator wins when the vertices he has\u0000claimed form a dominating set of the graph. Staller wins if she makes it\u0000impossible for Dominator to win, or equivalently, she is able to claim some\u0000vertex and all its neighbours. Maker-Breaker domination number $gamma_{MB}(G)$\u0000($gamma '_{MB}(G)$) of a graph $G$ is defined to be the minimum number of\u0000moves for Dominator to guarantee his winning when he plays first (second). We\u0000investigate these two invariants for the Cartesian product of any two graphs.\u0000We obtain upper bounds for the Maker-Breaker domination number of the Cartesian\u0000product of two arbitrary graphs. Also, we give upper bounds for the\u0000Maker-Breaker domination number of the Cartesian product of the complete graph\u0000with two vertices and an arbitrary graph. Most importantly, we prove that\u0000$gamma'_{MB}(P_2square P_n)=n$ for $ngeq 1$, $gamma_{MB}(P_2square P_n)$\u0000equals $n$, $n-1$, $n-2$, for $1leq nleq 4$, $5leq nleq 12$, and $ngeq\u000013$, respectively. For the disjoint union of $P_2square P_n$s, we show that\u0000$gamma_{MB}'(dotcup_{i=1}^k(P_2square P_n)_i)=kcdot n$ ($ngeq 1$), and\u0000that $gamma_{MB}(dotcup_{i=1}^k(P_2square P_n)_i)$ equals $kcdot n$,\u0000$kcdot n-1$, $kcdot n-2$ for $1leq nleq 4$, $5leq nleq 12$, and $ngeq\u000013$, respectively.","PeriodicalId":412397,"journal":{"name":"Discrete Mathematics & Theoretical Computer Science","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141209661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study a variant of graph domination known as $(t, r)$ broadcast domination, first defined in Blessing, Insko, Johnson, and Mauretour in 2015. In this variant, each broadcast provides $t-d$ reception to each vertex a distance $d < t$ from the broadcast. If $d ge t$ then no reception is provided. A vertex is considered dominated if it receives $r$ total reception from all broadcasts. Our main results provide some upper and lower bounds on the density of a $(t, r)$ dominating pattern of an infinite grid, as well as methods of computing them. Also, when $r ge 2$ we describe a family of counterexamples to a generalization of Vizing's Conjecture to $(t,r)$ broadcast domination.
{"title":"Bounds On $(t,r)$ Broadcast Domination of $n$-Dimensional Grids","authors":"T. Shlomi","doi":"10.46298/dmtcs.5732","DOIUrl":"https://doi.org/10.46298/dmtcs.5732","url":null,"abstract":"In this paper, we study a variant of graph domination known as $(t, r)$\u0000broadcast domination, first defined in Blessing, Insko, Johnson, and Mauretour\u0000in 2015. In this variant, each broadcast provides $t-d$ reception to each\u0000vertex a distance $d < t$ from the broadcast. If $d ge t$ then no reception is\u0000provided. A vertex is considered dominated if it receives $r$ total reception\u0000from all broadcasts. Our main results provide some upper and lower bounds on\u0000the density of a $(t, r)$ dominating pattern of an infinite grid, as well as\u0000methods of computing them. Also, when $r ge 2$ we describe a family of\u0000counterexamples to a generalization of Vizing's Conjecture to $(t,r)$ broadcast\u0000domination.","PeriodicalId":412397,"journal":{"name":"Discrete Mathematics & Theoretical Computer Science","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121138620","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the counting sequence for permutations avoiding both of the (classical) patterns 1243 and 2134 has the algebraic generating function supplied by Vaclav Kotesovec for sequence A164651 in The On-Line Encyclopedia of Integer Sequences.
{"title":"The number of {1243, 2134}-avoiding permutations","authors":"David Callan","doi":"10.46298/dmtcs.5287","DOIUrl":"https://doi.org/10.46298/dmtcs.5287","url":null,"abstract":"We show that the counting sequence for permutations avoiding both of the\u0000(classical) patterns 1243 and 2134 has the algebraic generating function\u0000supplied by Vaclav Kotesovec for sequence A164651 in The On-Line Encyclopedia\u0000of Integer Sequences.","PeriodicalId":412397,"journal":{"name":"Discrete Mathematics & Theoretical Computer Science","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124144125","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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