The study of minus paired domination of a graph G=(V,E) is initiated. Let SV be any paired dominating set of G , a minus paired dominating function is a function of the form f∶V→{-1,0,1} such that ...The study of minus paired domination of a graph G=(V,E) is initiated. Let SV be any paired dominating set of G , a minus paired dominating function is a function of the form f∶V→{-1,0,1} such that f(v)= 1 for v∈S, f(v)≤0 for v∈V-S , and f(N)≥1 for all v∈V . The weight of a minus paired dominating function f is w(f)=∑f(v) , over all vertices v∈V . The minus paired domination number of a graph G is γ - p( G )=min{ w(f)|f is a minus paired dominating function of G }. On the basis of the minus paired domination number of a graph G defined, some of its properties are discussed.展开更多
Let G=(V,E) be a simple graph. For any real valued function f:V →R, the weight of f is f(V) = ∑f(v) over all vertices v∈V . A signed total dominating function is a function f:V→{-1,1} such ...Let G=(V,E) be a simple graph. For any real valued function f:V →R, the weight of f is f(V) = ∑f(v) over all vertices v∈V . A signed total dominating function is a function f:V→{-1,1} such that f(N(v)) ≥1 for every vertex v∈V . The signed total domination number of a graph G equals the minimum weight of a signed total dominating function on G . In this paper, some properties of the signed total domination number of a graph G are discussed.展开更多
Let G = (V, E) be a graph, and let f : V →{-1, 1} be a two-valued function. If ∑x∈N(v) f(x) ≥ 1 for each v ∈ V, where N(v) is the open neighborhood of v, then f is a signed total dominating function on ...Let G = (V, E) be a graph, and let f : V →{-1, 1} be a two-valued function. If ∑x∈N(v) f(x) ≥ 1 for each v ∈ V, where N(v) is the open neighborhood of v, then f is a signed total dominating function on G. A set {fl, f2,… fd} of signed d total dominating functions on G with the property that ∑i=1^d fi(x) ≤ 1 for each x ∈ V, is called a signed total dominating family (of functions) on G. The maximum number of functions in a signed total dominating family on G is the signed total domatic number on G, denoted by dt^s(G). The properties of the signed total domatic number dt^s(G) are studied in this paper. In particular, we give the sharp bounds of the signed total domatic number of regular graphs, complete bipartite graphs and complete graphs.展开更多
Let <em>G</em>(<em>V</em>, <em>E</em>) be a finite connected simple graph with vertex set <em>V</em>(<em>G</em>). A function is a signed dominating function ...Let <em>G</em>(<em>V</em>, <em>E</em>) be a finite connected simple graph with vertex set <em>V</em>(<em>G</em>). A function is a signed dominating function <em>f </em>: <em style="white-space:normal;">V</em><span style="white-space:normal;">(</span><em style="white-space:normal;">G</em><span style="white-space:normal;">)</span><span style="white-space:nowrap;">→{<span style="white-space:nowrap;"><span style="white-space:nowrap;">−</span></span>1,1}</span> if for every vertex <em>v</em> <span style="white-space:nowrap;">∈</span> <em>V</em>(<em>G</em>), the sum of closed neighborhood weights of <em>v</em> is greater or equal to 1. The signed domination number <em>γ</em><sub>s</sub>(<em>G</em>) of <em>G</em> is the minimum weight of a signed dominating function on <em>G</em>. In this paper, we calculate the signed domination numbers of the Cartesian product of two paths <em>P</em><sub><em>m</em></sub> and <em>P</em><sub><em>n</em></sub> for <em>m</em> = 6, 7 and arbitrary <em>n</em>.展开更多
Let <img src="Edit_092a0db1-eefa-4bff-81a0-751d038158ad.png" width="58" height="20" alt="" /> be a graph. A function <img src="Edit_b7158ed5-6825-41cd-b7f0-5ab5e16...Let <img src="Edit_092a0db1-eefa-4bff-81a0-751d038158ad.png" width="58" height="20" alt="" /> be a graph. A function <img src="Edit_b7158ed5-6825-41cd-b7f0-5ab5e16fc53d.png" width="79" height="20" alt="" /> is said to be a Signed Dominating Function (SDF) if <img src="Edit_c6e63805-bcaa-46a9-bc77-42750af8efd4.png" width="135" height="25" alt="" /> holds for all <img src="Edit_bba1b366-af70-46cd-aefe-fc68869da670.png" width="42" height="20" alt="" />. The signed domination number <img src="Edit_22e6d87a-e3be-4037-b4b6-c1de6a40abb0.png" width="284" height="25" alt="" />. In this paper, we determine the exact value of the Signed Domination Number of graphs <img src="Edit_36ef2747-da44-4f9b-a10a-340c61a3f28c.png" width="19" height="20" alt="" /> and <img src="Edit_26eb0f74-fcc2-49ad-8567-492cf3115b73.png" width="19" height="20" alt="" /> for <img src="Edit_856dbcc1-d215-4144-b50c-ac8a225d664f.png" width="32" height="20" alt="" />, which is generalized the known results, respectively, where <img src="Edit_4b7e4f8f-5d38-4fd0-ac4e-dd8ef243029f.png" width="19" height="20" alt="" /> and <img src="Edit_6557afba-e697-4397-994e-a9bda83e3219.png" width="19" height="20" alt="" /> are denotes the <em>k</em>-th power graphs of cycle <img src="Edit_27e6e80f-85d5-4208-b367-a757a0e55d0b.png" width="21" height="20" alt="" /> and path <img src="Edit_70ac5266-950b-4bfd-8d04-21711d3ffc33.png" width="18" height="20" alt="" />.展开更多
In this paper, we introduce the concepts of redundant constraint and exceptional vertex which play an important role in the characterization of universal minimal total dominating functions (universal MTDFs), and estab...In this paper, we introduce the concepts of redundant constraint and exceptional vertex which play an important role in the characterization of universal minimal total dominating functions (universal MTDFs), and establish some further results on universal MTDFs in general graphs. By extending these results to trees, we get a necessary and sufficient condition for universal MTDFs and show that there is a good algorithm for deciding whether a given tree has a universal MTDF.展开更多
Let G = (V,E) be a graph.A function f : E → {-1,1} is said to be a signed edge total dominating function (SETDF) of G if e ∈N(e) f(e ) ≥ 1 holds for every edge e ∈ E(G).The signed edge total domination ...Let G = (V,E) be a graph.A function f : E → {-1,1} is said to be a signed edge total dominating function (SETDF) of G if e ∈N(e) f(e ) ≥ 1 holds for every edge e ∈ E(G).The signed edge total domination number γ st (G) of G is defined as γ st (G) = min{ e∈E(G) f(e)|f is an SETDF of G}.In this paper we obtain some new lower bounds of γ st (G).展开更多
Let G = (V, E) be a simple graph. A function f : E → {+1,-1} is called a signed cycle domination function (SCDF) of G if ∑e∈E(C) f(e) ≥ 1 for every induced cycle C of G. The signed cycle domination numbe...Let G = (V, E) be a simple graph. A function f : E → {+1,-1} is called a signed cycle domination function (SCDF) of G if ∑e∈E(C) f(e) ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γ′sc(G) = min{∑e∈E f(e)| f is an SCDF of G}. This paper will characterize all maxima] planar graphs G with order n ≥ 6 and γ′sc(G) =n.展开更多
Let G=(V,E)be a graph,for an element x∈V∪E,the open total neighborhood of x is denoted by N_(t)(x)={y|y is adjacent to x or y is incident with x,y∈V∪E},and Nt[x]=Nt(x)∪{x}is the closed one.A function f:V(G)∪E(G...Let G=(V,E)be a graph,for an element x∈V∪E,the open total neighborhood of x is denoted by N_(t)(x)={y|y is adjacent to x or y is incident with x,y∈V∪E},and Nt[x]=Nt(x)∪{x}is the closed one.A function f:V(G)∪E(G)→{−1,0,1}is said to be a mixed minus domination function(TMDF)of G if∑_(y∈Nt[x])f(y)≥1 holds for all x∈V(G)∪E(G).The mixed minus domination numberγ′_(tm)(G)of G is defined as γ′_(tm)(G)=min{∑x∈V∪E f(x)|f is a TMDF of G.In this paper,we obtain some lower bounds of the mixed minus domination number of G and give the exact values ofγ′_(tm)(G)when G is a cycle or a path.展开更多
文摘The study of minus paired domination of a graph G=(V,E) is initiated. Let SV be any paired dominating set of G , a minus paired dominating function is a function of the form f∶V→{-1,0,1} such that f(v)= 1 for v∈S, f(v)≤0 for v∈V-S , and f(N)≥1 for all v∈V . The weight of a minus paired dominating function f is w(f)=∑f(v) , over all vertices v∈V . The minus paired domination number of a graph G is γ - p( G )=min{ w(f)|f is a minus paired dominating function of G }. On the basis of the minus paired domination number of a graph G defined, some of its properties are discussed.
文摘Let G=(V,E) be a simple graph. For any real valued function f:V →R, the weight of f is f(V) = ∑f(v) over all vertices v∈V . A signed total dominating function is a function f:V→{-1,1} such that f(N(v)) ≥1 for every vertex v∈V . The signed total domination number of a graph G equals the minimum weight of a signed total dominating function on G . In this paper, some properties of the signed total domination number of a graph G are discussed.
基金Project supported by the National Natural Science Foundation of China (Grant No.1057117), and the Science Foundation of Shanghai Municipal Commission of Education (Grant No.05AZ04).
文摘Let G = (V, E) be a graph, and let f : V →{-1, 1} be a two-valued function. If ∑x∈N(v) f(x) ≥ 1 for each v ∈ V, where N(v) is the open neighborhood of v, then f is a signed total dominating function on G. A set {fl, f2,… fd} of signed d total dominating functions on G with the property that ∑i=1^d fi(x) ≤ 1 for each x ∈ V, is called a signed total dominating family (of functions) on G. The maximum number of functions in a signed total dominating family on G is the signed total domatic number on G, denoted by dt^s(G). The properties of the signed total domatic number dt^s(G) are studied in this paper. In particular, we give the sharp bounds of the signed total domatic number of regular graphs, complete bipartite graphs and complete graphs.
文摘Let <em>G</em>(<em>V</em>, <em>E</em>) be a finite connected simple graph with vertex set <em>V</em>(<em>G</em>). A function is a signed dominating function <em>f </em>: <em style="white-space:normal;">V</em><span style="white-space:normal;">(</span><em style="white-space:normal;">G</em><span style="white-space:normal;">)</span><span style="white-space:nowrap;">→{<span style="white-space:nowrap;"><span style="white-space:nowrap;">−</span></span>1,1}</span> if for every vertex <em>v</em> <span style="white-space:nowrap;">∈</span> <em>V</em>(<em>G</em>), the sum of closed neighborhood weights of <em>v</em> is greater or equal to 1. The signed domination number <em>γ</em><sub>s</sub>(<em>G</em>) of <em>G</em> is the minimum weight of a signed dominating function on <em>G</em>. In this paper, we calculate the signed domination numbers of the Cartesian product of two paths <em>P</em><sub><em>m</em></sub> and <em>P</em><sub><em>n</em></sub> for <em>m</em> = 6, 7 and arbitrary <em>n</em>.
文摘Let <img src="Edit_092a0db1-eefa-4bff-81a0-751d038158ad.png" width="58" height="20" alt="" /> be a graph. A function <img src="Edit_b7158ed5-6825-41cd-b7f0-5ab5e16fc53d.png" width="79" height="20" alt="" /> is said to be a Signed Dominating Function (SDF) if <img src="Edit_c6e63805-bcaa-46a9-bc77-42750af8efd4.png" width="135" height="25" alt="" /> holds for all <img src="Edit_bba1b366-af70-46cd-aefe-fc68869da670.png" width="42" height="20" alt="" />. The signed domination number <img src="Edit_22e6d87a-e3be-4037-b4b6-c1de6a40abb0.png" width="284" height="25" alt="" />. In this paper, we determine the exact value of the Signed Domination Number of graphs <img src="Edit_36ef2747-da44-4f9b-a10a-340c61a3f28c.png" width="19" height="20" alt="" /> and <img src="Edit_26eb0f74-fcc2-49ad-8567-492cf3115b73.png" width="19" height="20" alt="" /> for <img src="Edit_856dbcc1-d215-4144-b50c-ac8a225d664f.png" width="32" height="20" alt="" />, which is generalized the known results, respectively, where <img src="Edit_4b7e4f8f-5d38-4fd0-ac4e-dd8ef243029f.png" width="19" height="20" alt="" /> and <img src="Edit_6557afba-e697-4397-994e-a9bda83e3219.png" width="19" height="20" alt="" /> are denotes the <em>k</em>-th power graphs of cycle <img src="Edit_27e6e80f-85d5-4208-b367-a757a0e55d0b.png" width="21" height="20" alt="" /> and path <img src="Edit_70ac5266-950b-4bfd-8d04-21711d3ffc33.png" width="18" height="20" alt="" />.
文摘In this paper, we introduce the concepts of redundant constraint and exceptional vertex which play an important role in the characterization of universal minimal total dominating functions (universal MTDFs), and establish some further results on universal MTDFs in general graphs. By extending these results to trees, we get a necessary and sufficient condition for universal MTDFs and show that there is a good algorithm for deciding whether a given tree has a universal MTDF.
基金Supported by the National Natural Science Foundation of China (Grant No. 11061014)
文摘Let G = (V,E) be a graph.A function f : E → {-1,1} is said to be a signed edge total dominating function (SETDF) of G if e ∈N(e) f(e ) ≥ 1 holds for every edge e ∈ E(G).The signed edge total domination number γ st (G) of G is defined as γ st (G) = min{ e∈E(G) f(e)|f is an SETDF of G}.In this paper we obtain some new lower bounds of γ st (G).
基金Supported by Doctoral Scientific Research Fund of Harbin Normal University(Grant No.KGB201008)
文摘Let G = (V, E) be a simple graph. A function f : E → {+1,-1} is called a signed cycle domination function (SCDF) of G if ∑e∈E(C) f(e) ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γ′sc(G) = min{∑e∈E f(e)| f is an SCDF of G}. This paper will characterize all maxima] planar graphs G with order n ≥ 6 and γ′sc(G) =n.
基金This work was supported by the National Natural Science Foundation of China(No.11061014,11361024,11261019)the Science Foundation of Jiangxi Province(No.KJLD12067)The authors are grateful to the referees for their careful reading with corrections and especially the referee who draws our attention to the proof in Theorem 2.2,which let us improve the proof of Theorem 2.2,and correct this lower bound.
文摘Let G=(V,E)be a graph,for an element x∈V∪E,the open total neighborhood of x is denoted by N_(t)(x)={y|y is adjacent to x or y is incident with x,y∈V∪E},and Nt[x]=Nt(x)∪{x}is the closed one.A function f:V(G)∪E(G)→{−1,0,1}is said to be a mixed minus domination function(TMDF)of G if∑_(y∈Nt[x])f(y)≥1 holds for all x∈V(G)∪E(G).The mixed minus domination numberγ′_(tm)(G)of G is defined as γ′_(tm)(G)=min{∑x∈V∪E f(x)|f is a TMDF of G.In this paper,we obtain some lower bounds of the mixed minus domination number of G and give the exact values ofγ′_(tm)(G)when G is a cycle or a path.