Pub Date : 2019-01-01DOI: 10.20429/TAG.2019.060102
Kwalah Alhulwah, M. Zayed, Ping Zhang
One of the most familiar derived graphs is the line graph. The line graph L(G) of a graph G is that graph whose vertices are the edges of G where two vertices of L(G) are adjacent if the corresponding edges are adjacent in G. Two nontrivial paths P and Q in a graph G are said to be adjacent paths in G if P and Q have exactly one vertex in common and this vertex is an end-vertex of both P and Q. For an integer l ≥ 2, the l-line graph Ll(G) of a graph G is the graph whose vertex set is the set of all l-paths (paths of order l) of G where two vertices of Ll(G) are adjacent if they are adjacent l-paths in G. Since the 2-line graph is the line graph L(G) for every graph G, this is a generalization of line graphs. In this work, we study planar and outerplanar properties of the 3-line graph of connected graphs and present characterizations of those trees having a planar or outerplanar 3-line graph by means of forbidden subtrees.
{"title":"On the Planarity of Generalized Line Graphs","authors":"Kwalah Alhulwah, M. Zayed, Ping Zhang","doi":"10.20429/TAG.2019.060102","DOIUrl":"https://doi.org/10.20429/TAG.2019.060102","url":null,"abstract":"One of the most familiar derived graphs is the line graph. The line graph L(G) of a graph G is that graph whose vertices are the edges of G where two vertices of L(G) are adjacent if the corresponding edges are adjacent in G. Two nontrivial paths P and Q in a graph G are said to be adjacent paths in G if P and Q have exactly one vertex in common and this vertex is an end-vertex of both P and Q. For an integer l ≥ 2, the l-line graph Ll(G) of a graph G is the graph whose vertex set is the set of all l-paths (paths of order l) of G where two vertices of Ll(G) are adjacent if they are adjacent l-paths in G. Since the 2-line graph is the line graph L(G) for every graph G, this is a generalization of line graphs. In this work, we study planar and outerplanar properties of the 3-line graph of connected graphs and present characterizations of those trees having a planar or outerplanar 3-line graph by means of forbidden subtrees.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67545239","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 : 2019-01-01DOI: 10.20429/tag.2019.060205
Mohamad Abdallah, E. Cheng
The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. Park and Ihm introduced the problem of strong matching preclusion under the condition that no isolated vertex is created as a result of faults. In this paper, we find the conditional strong matching preclusion number for the n-dimensional alternating group graph AGn.
{"title":"Conditional Strong Matching Preclusion of the Alternating Group Graph","authors":"Mohamad Abdallah, E. Cheng","doi":"10.20429/tag.2019.060205","DOIUrl":"https://doi.org/10.20429/tag.2019.060205","url":null,"abstract":"The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. Park and Ihm introduced the problem of strong matching preclusion under the condition that no isolated vertex is created as a result of faults. In this paper, we find the conditional strong matching preclusion number for the n-dimensional alternating group graph AGn.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67545658","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 : 2018-04-24DOI: 10.20429/tag.2021.080102
D. Basnet, Ajay Sharma, Rahul Dutta
In this article, we introduce the concept of nilpotent graph of a finite commutative ring. The set of all non nilpotent elements of a ring is taken as the vertex set and two vertices are adjacent if and only if their sum is nilpotent. We discuss some graph theoretic properties of nilpotent graph.
{"title":"Nilpotent Graph","authors":"D. Basnet, Ajay Sharma, Rahul Dutta","doi":"10.20429/tag.2021.080102","DOIUrl":"https://doi.org/10.20429/tag.2021.080102","url":null,"abstract":"In this article, we introduce the concept of nilpotent graph of a finite commutative ring. The set of all non nilpotent elements of a ring is taken as the vertex set and two vertices are adjacent if and only if their sum is nilpotent. We discuss some graph theoretic properties of nilpotent graph.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43976553","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 : 2017-09-21DOI: 10.20429/TAG.2019.060106
Y. Caro, R. Pepper
The maximum oriented $k$-forcing number of a simple graph $G$, written $MOF_k(G)$, is the maximum directed $k$-forcing number among all orientations of $G$. This invariant was recently introduced by Caro, Davila and Pepper in [CaroDavilaPepper], and in the current paper we study the special case where $G$ is the complete graph with order $n$, denoted $K_n$. While $MOF_k(G)$ is an invariant for the underlying simple graph $G$, $MOF_k(K_n)$ can also be interpreted as an interesting property for tournaments. Our main results further focus on the case when $k=1$. These include a lower bound on $MOF(K_n)$ of roughly $frac{3}{4}n$, and for $nge 2$, a lower bound of $n - frac{2n}{log_2(n)}$. Along the way, we also consider various lower bounds on the maximum oriented $k$-forcing number for the closely related complete $q$-partite graphs.
{"title":"Maximum Oriented Forcing Number for Complete Graphs","authors":"Y. Caro, R. Pepper","doi":"10.20429/TAG.2019.060106","DOIUrl":"https://doi.org/10.20429/TAG.2019.060106","url":null,"abstract":"The maximum oriented $k$-forcing number of a simple graph $G$, written $MOF_k(G)$, is the maximum directed $k$-forcing number among all orientations of $G$. This invariant was recently introduced by Caro, Davila and Pepper in [CaroDavilaPepper], and in the current paper we study the special case where $G$ is the complete graph with order $n$, denoted $K_n$. While $MOF_k(G)$ is an invariant for the underlying simple graph $G$, $MOF_k(K_n)$ can also be interpreted as an interesting property for tournaments. Our main results further focus on the case when $k=1$. These include a lower bound on $MOF(K_n)$ of roughly $frac{3}{4}n$, and for $nge 2$, a lower bound of $n - frac{2n}{log_2(n)}$. Along the way, we also consider various lower bounds on the maximum oriented $k$-forcing number for the closely related complete $q$-partite graphs.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44617828","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 : 2017-08-04DOI: 10.20429/TAG.2020.070104
Randy Davila, E. Krop
Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $gamma(G)$. For any integer $kge 1$, a function $f : V (G) rightarrow {0, 1, . . ., k}$ is called a emph{${k}$-dominating function} if the sum of its function values over any closed neighborhood is at least $k$. The weight of a ${k}$-dominating function is the sum of its values over all the vertices. The ${k}$-domination number of $G$, $gamma_{{k}}(G)$, is defined to be the minimum weight taken over all ${k}$-domination functions. Bresar, Henning, and Klavžar (On integer domination in graphs and Vizing-like problems. emph{Taiwanese J. Math.} {10(5)} (2006) pp. 1317--1328) asked whether there exists an integer $kge 2$ so that $gamma_{{k}}(Gsquare H)ge gamma(G)gamma(H)$. In this note we use the Roman ${2}$-domination number, $gamma_{R2}$ of Chellali, Haynes, Hedetniemi, and McRae, (Roman ${2}$-domination. emph{Discrete Applied Mathematics} {204} (2016) pp. 22-28.) to prove that if $G$ is a claw-free graph and $H$ is an arbitrary graph, then $gamma_{{2}}(Gsquare H)ge gamma_{R2}(Gsquare H)ge gamma(G)gamma(H)$, which also implies the conjecture for all $kge 2$.
{"title":"On a Vizing-type Integer Domination Conjecture","authors":"Randy Davila, E. Krop","doi":"10.20429/TAG.2020.070104","DOIUrl":"https://doi.org/10.20429/TAG.2020.070104","url":null,"abstract":"Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $gamma(G)$. For any integer $kge 1$, a function $f : V (G) rightarrow {0, 1, . . ., k}$ is called a emph{${k}$-dominating function} if the sum of its function values over any closed neighborhood is at least $k$. The weight of a ${k}$-dominating function is the sum of its values over all the vertices. The ${k}$-domination number of $G$, $gamma_{{k}}(G)$, is defined to be the minimum weight taken over all ${k}$-domination functions. Bresar, Henning, and Klavžar (On integer domination in graphs and Vizing-like problems. emph{Taiwanese J. Math.} {10(5)} (2006) pp. 1317--1328) asked whether there exists an integer $kge 2$ so that $gamma_{{k}}(Gsquare H)ge gamma(G)gamma(H)$. In this note we use the Roman ${2}$-domination number, $gamma_{R2}$ of Chellali, Haynes, Hedetniemi, and McRae, (Roman ${2}$-domination. emph{Discrete Applied Mathematics} {204} (2016) pp. 22-28.) to prove that if $G$ is a claw-free graph and $H$ is an arbitrary graph, then $gamma_{{2}}(Gsquare H)ge gamma_{R2}(Gsquare H)ge gamma(G)gamma(H)$, which also implies the conjecture for all $kge 2$.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42146416","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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