Many graph domination applications can be expanded to achieve complete cototal domination.If every node in a dominating set is regarded as a record server for a PC organization,then each PC affiliated with the organiz...Many graph domination applications can be expanded to achieve complete cototal domination.If every node in a dominating set is regarded as a record server for a PC organization,then each PC affiliated with the organization has direct access to a document server.It is occasionally reasonable to believe that this gateway will remain available even if one of the scrape servers fails.Because every PC has direct access to at least two documents’servers,a complete cototal dominating set provides the required adaptability to non-critical failure in such scenarios.In this paper,we presented a method for calculating a graph’s complete cototal roman domination number.We also examined the properties and determined the bounds for a graph’s complete cototal roman domination number,and its applications are presented.It has been observed that one’s interest fluctuate over time,therefore inferring them just from one’s own behaviour may be inconclusive.However,it may be able to deduce a user’s constant interest to some level if a user’s networking is also watched for similar or related actions.This research proposes a method that considers a user’s and his channel’s activity,as well as common tags,persons,and organizations from their social media posts in order to establish a solid foundation for the required conclusion.展开更多
A signed(res. signed total) Roman dominating function, SRDF(res.STRDF) for short, of a graph G =(V, E) is a function f : V → {-1, 1, 2} satisfying the conditions that(i)∑v∈N[v]f(v) ≥ 1(res.∑v∈N(v)f(v) ≥ 1) for ...A signed(res. signed total) Roman dominating function, SRDF(res.STRDF) for short, of a graph G =(V, E) is a function f : V → {-1, 1, 2} satisfying the conditions that(i)∑v∈N[v]f(v) ≥ 1(res.∑v∈N(v)f(v) ≥ 1) for any v ∈ V, where N [v] is the closed neighborhood and N(v) is the neighborhood of v, and(ii) every vertex v for which f(v) =-1 is adjacent to a vertex u for which f(u) = 2. The weight of a SRDF(res. STRDF) is the sum of its function values over all vertices.The signed(res. signed total) Roman domination number of G is the minimum weight among all signed(res. signed total) Roman dominating functions of G. In this paper,we compute the exact values of the signed(res. signed total) Roman domination numbers of complete bipartite graphs and wheels.展开更多
Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f ： V（G） → {0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f1, f2,..., fd} of disti...Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f ： V（G） → {0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f1, f2,..., fd} of distinct Roman k-dominating functions on G with the property that ∑di=1 fi（v） ≤ 2 for each v C V（G）, is called a Roman k-dominating family （of functions） on G. The maximum number of functions in a Roman k-dominating family on G is the Roman k-domatic number of G, denoted by dkR（G）. Note that the Roman 1-domatic number dlR（G） is the usual Roman domatic number dR（G）. In this paper we initiate the study of the Roman k-domatic number in graphs and we present sharp bounds for dkR（G）. In addition, we determine the Roman k-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman domatic number.展开更多
文摘Many graph domination applications can be expanded to achieve complete cototal domination.If every node in a dominating set is regarded as a record server for a PC organization,then each PC affiliated with the organization has direct access to a document server.It is occasionally reasonable to believe that this gateway will remain available even if one of the scrape servers fails.Because every PC has direct access to at least two documents’servers,a complete cototal dominating set provides the required adaptability to non-critical failure in such scenarios.In this paper,we presented a method for calculating a graph’s complete cototal roman domination number.We also examined the properties and determined the bounds for a graph’s complete cototal roman domination number,and its applications are presented.It has been observed that one’s interest fluctuate over time,therefore inferring them just from one’s own behaviour may be inconclusive.However,it may be able to deduce a user’s constant interest to some level if a user’s networking is also watched for similar or related actions.This research proposes a method that considers a user’s and his channel’s activity,as well as common tags,persons,and organizations from their social media posts in order to establish a solid foundation for the required conclusion.
基金The NSF(11271365)of Chinathe NSF(BK20151117)of Jiangsu Province
文摘A signed(res. signed total) Roman dominating function, SRDF(res.STRDF) for short, of a graph G =(V, E) is a function f : V → {-1, 1, 2} satisfying the conditions that(i)∑v∈N[v]f(v) ≥ 1(res.∑v∈N(v)f(v) ≥ 1) for any v ∈ V, where N [v] is the closed neighborhood and N(v) is the neighborhood of v, and(ii) every vertex v for which f(v) =-1 is adjacent to a vertex u for which f(u) = 2. The weight of a SRDF(res. STRDF) is the sum of its function values over all vertices.The signed(res. signed total) Roman domination number of G is the minimum weight among all signed(res. signed total) Roman dominating functions of G. In this paper,we compute the exact values of the signed(res. signed total) Roman domination numbers of complete bipartite graphs and wheels.
文摘Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f ： V（G） → {0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f1, f2,..., fd} of distinct Roman k-dominating functions on G with the property that ∑di=1 fi（v） ≤ 2 for each v C V（G）, is called a Roman k-dominating family （of functions） on G. The maximum number of functions in a Roman k-dominating family on G is the Roman k-domatic number of G, denoted by dkR（G）. Note that the Roman 1-domatic number dlR（G） is the usual Roman domatic number dR（G）. In this paper we initiate the study of the Roman k-domatic number in graphs and we present sharp bounds for dkR（G）. In addition, we determine the Roman k-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman domatic number.
