Pub Date : 2023-08-11DOI: 10.1142/s1793830923500751
Shivani Chauhan, A. Satyanarayana Reddy
Let $X$ be a tree with $n$ vertices and $L(X)$ be its line graph. In this work, we completely characterize the trees for which the algebraic connectivity of $L(X)times K_m$ is equal to $m-1$, where $times$ denotes the Kronecker product. We provide a few necessary and sufficient conditions for $L(X)times K_m$ to be Laplacian integral. The algebraic connectivity of $L(X)times K_m$, where $X$ is a tree of diameter $4$ and $k$-book graph is discussed.
{"title":"Algebraic connectivity of Kronecker products of line graphs","authors":"Shivani Chauhan, A. Satyanarayana Reddy","doi":"10.1142/s1793830923500751","DOIUrl":"https://doi.org/10.1142/s1793830923500751","url":null,"abstract":"Let $X$ be a tree with $n$ vertices and $L(X)$ be its line graph. In this work, we completely characterize the trees for which the algebraic connectivity of $L(X)times K_m$ is equal to $m-1$, where $times$ denotes the Kronecker product. We provide a few necessary and sufficient conditions for $L(X)times K_m$ to be Laplacian integral. The algebraic connectivity of $L(X)times K_m$, where $X$ is a tree of diameter $4$ and $k$-book graph is discussed.","PeriodicalId":45568,"journal":{"name":"Discrete Mathematics Algorithms and Applications","volume":"24 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78708511","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}
Pub Date : 2023-08-10DOI: 10.1142/s1793830923500696
V. Swaminathan, R. Sundareswaran, L. Muthusubramanian
{"title":"Locating fair domination in graphs","authors":"V. Swaminathan, R. Sundareswaran, L. Muthusubramanian","doi":"10.1142/s1793830923500696","DOIUrl":"https://doi.org/10.1142/s1793830923500696","url":null,"abstract":"","PeriodicalId":45568,"journal":{"name":"Discrete Mathematics Algorithms and Applications","volume":"106 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80922429","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}
Pub Date : 2023-08-04DOI: 10.1142/s1793830923500684
P. R. L. Pushpam, M. Kamalam, B. Mahavir
{"title":"Stability of Certified Domination upon Edge Addition","authors":"P. R. L. Pushpam, M. Kamalam, B. Mahavir","doi":"10.1142/s1793830923500684","DOIUrl":"https://doi.org/10.1142/s1793830923500684","url":null,"abstract":"","PeriodicalId":45568,"journal":{"name":"Discrete Mathematics Algorithms and Applications","volume":"27 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78584828","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}
Pub Date : 2023-07-28DOI: 10.1142/s1793830923500647
T. Kavaskar, K. Vinothkumar
{"title":"Padmakar-Ivan Index of Some Families of Graphs","authors":"T. Kavaskar, K. Vinothkumar","doi":"10.1142/s1793830923500647","DOIUrl":"https://doi.org/10.1142/s1793830923500647","url":null,"abstract":"","PeriodicalId":45568,"journal":{"name":"Discrete Mathematics Algorithms and Applications","volume":"44 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84684922","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}
The total domination game is played on a simple graph [Formula: see text] with no isolated vertices by two players, Dominator and Staller, who alternate choosing a vertex in [Formula: see text]. Each chosen vertex totally dominates its neighbors. In this game, each chosen vertex must totally dominates at least one new vertex not totally dominated before. The game ends when all vertices in [Formula: see text] are totally dominated. Dominator’s goal is to finish the game as soon as possible, and Staller’s goal is to prolong it as much as possible. The game total domination number is the number of chosen vertices when both players play optimally, denoted by [Formula: see text] when Dominator starts the game and denoted by [Formula: see text] when Staller starts the game. If a vertex [Formula: see text] in [Formula: see text] is declared to be already totally dominated, then we denote this graph by [Formula: see text]. A total domination game critical graph is a graph [Formula: see text] for which [Formula: see text] holds for every vertex [Formula: see text] in [Formula: see text]. If [Formula: see text], then [Formula: see text] is called [Formula: see text]-[Formula: see text]-critical. In this work, we characterize some 4-[Formula: see text]-critical graphs.
{"title":"4-Total domination game critical graphs","authors":"Chalermpong Worawannotai, Karnchana Charoensitthichai","doi":"10.1142/s1793830923500611","DOIUrl":"https://doi.org/10.1142/s1793830923500611","url":null,"abstract":"The total domination game is played on a simple graph [Formula: see text] with no isolated vertices by two players, Dominator and Staller, who alternate choosing a vertex in [Formula: see text]. Each chosen vertex totally dominates its neighbors. In this game, each chosen vertex must totally dominates at least one new vertex not totally dominated before. The game ends when all vertices in [Formula: see text] are totally dominated. Dominator’s goal is to finish the game as soon as possible, and Staller’s goal is to prolong it as much as possible. The game total domination number is the number of chosen vertices when both players play optimally, denoted by [Formula: see text] when Dominator starts the game and denoted by [Formula: see text] when Staller starts the game. If a vertex [Formula: see text] in [Formula: see text] is declared to be already totally dominated, then we denote this graph by [Formula: see text]. A total domination game critical graph is a graph [Formula: see text] for which [Formula: see text] holds for every vertex [Formula: see text] in [Formula: see text]. If [Formula: see text], then [Formula: see text] is called [Formula: see text]-[Formula: see text]-critical. In this work, we characterize some 4-[Formula: see text]-critical graphs.","PeriodicalId":45568,"journal":{"name":"Discrete Mathematics Algorithms and Applications","volume":"33 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87865777","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}
Pub Date : 2023-07-17DOI: 10.1142/s1793830923500581
Y. Egawa, Kenjiro Ogawa, K. Ozeki, Satoshi Tagusari, M. Tsuchiya
For a poset [Formula: see text] the strict-double-bound graph ([Formula: see text]) of [Formula: see text] is the graph with the vertex set [Formula: see text] such that [Formula: see text] if and only if [Formula: see text] and there exist [Formula: see text] and [Formula: see text] distinct from [Formula: see text] and [Formula: see text] such that [Formula: see text] and [Formula: see text] For a connected graph [Formula: see text], the strict-double-bound number [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the graph with [Formula: see text] vertices and no edges. In this paper we deal with the strict-double-bound numbers of Cartesian products of graphs. We show that [Formula: see text] for [Formula: see text], [Formula: see text] for [Formula: see text], and [Formula: see text] for [Formula: see text].
对于一个偏序集[公式:看到文本]strict-double-bound图([公式:看到文本])(公式:看到文本)是图的顶点集(公式:看到文本),(公式:看到文本)当且仅当(公式:看到文本)和存在[公式:看到文本]和[公式:看到文本)不同于[公式:看到文本]和[公式:看到文本),[公式:看到文本]和[公式:看到文本)的连通图(公式:看到文本),strict-double-bound数量(公式:定义[Formula: see text]为[Formula: see text]，其中[Formula: see text]为有[Formula: see text]顶点而无边的图形。本文研究了图的笛卡尔积的严格双界数。我们展示了[公式:见文本]对应[公式:见文本]，[公式:见文本]对应[公式:见文本]，[公式:见文本]对应[公式:见文本]。
{"title":"On strict-double-bound graphs and Cartesian products of paths and cycles","authors":"Y. Egawa, Kenjiro Ogawa, K. Ozeki, Satoshi Tagusari, M. Tsuchiya","doi":"10.1142/s1793830923500581","DOIUrl":"https://doi.org/10.1142/s1793830923500581","url":null,"abstract":"For a poset [Formula: see text] the strict-double-bound graph ([Formula: see text]) of [Formula: see text] is the graph with the vertex set [Formula: see text] such that [Formula: see text] if and only if [Formula: see text] and there exist [Formula: see text] and [Formula: see text] distinct from [Formula: see text] and [Formula: see text] such that [Formula: see text] and [Formula: see text] For a connected graph [Formula: see text], the strict-double-bound number [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the graph with [Formula: see text] vertices and no edges. In this paper we deal with the strict-double-bound numbers of Cartesian products of graphs. We show that [Formula: see text] for [Formula: see text], [Formula: see text] for [Formula: see text], and [Formula: see text] for [Formula: see text].","PeriodicalId":45568,"journal":{"name":"Discrete Mathematics Algorithms and Applications","volume":"36 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87307732","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}
Pub Date : 2023-07-13DOI: 10.1142/s1793830923500635
Parveen, Sandeep Dalal, J. Kumar
{"title":"Enhanced Power Graphs of Certain Non-Abelian Groups","authors":"Parveen, Sandeep Dalal, J. Kumar","doi":"10.1142/s1793830923500635","DOIUrl":"https://doi.org/10.1142/s1793830923500635","url":null,"abstract":"","PeriodicalId":45568,"journal":{"name":"Discrete Mathematics Algorithms and Applications","volume":"47 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78399030","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}
Pub Date : 2023-07-12DOI: 10.1142/s179383092350060x
Kijung Kim
The middle graph [Formula: see text] of a graph [Formula: see text] is the graph obtained by subdividing each edge of [Formula: see text] exactly once and joining all these newly introduced vertices of adjacent edges of [Formula: see text]. It is known that the decision problems for Italian domination number [Formula: see text], [Formula: see text]-rainbow domination number [Formula: see text] and Roman domination number [Formula: see text] are NP-complete. Recently, it was proved that [Formula: see text] for a graph [Formula: see text] of order [Formula: see text]. In this paper, we continue to study several domination and domatic numbers in middle graphs. We also obtain closed formulas for domination, secure domination and total domination numbers in middle graphs of rooted product graphs.
{"title":"Domination-related parameters in middle graphs","authors":"Kijung Kim","doi":"10.1142/s179383092350060x","DOIUrl":"https://doi.org/10.1142/s179383092350060x","url":null,"abstract":"The middle graph [Formula: see text] of a graph [Formula: see text] is the graph obtained by subdividing each edge of [Formula: see text] exactly once and joining all these newly introduced vertices of adjacent edges of [Formula: see text]. It is known that the decision problems for Italian domination number [Formula: see text], [Formula: see text]-rainbow domination number [Formula: see text] and Roman domination number [Formula: see text] are NP-complete. Recently, it was proved that [Formula: see text] for a graph [Formula: see text] of order [Formula: see text]. In this paper, we continue to study several domination and domatic numbers in middle graphs. We also obtain closed formulas for domination, secure domination and total domination numbers in middle graphs of rooted product graphs.","PeriodicalId":45568,"journal":{"name":"Discrete Mathematics Algorithms and Applications","volume":"42 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75820469","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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