Mathematics Department. School of Arts and Sciences. University of San Carlos, Cebu City. October 24-26, 2017. E.S.-E. and E.L.E.. Convex Secure Domination ...
Convex Secure Domination in Graphs Under Some Binary Operations Evelyn Samper-Enriquez and Enrico L. Enriquez Mathematics Department School of Arts and Sciences University of San Carlos, Cebu City
October 24-26, 2017
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Convex Secure Domination in Graphs Under Some B
Theoretical Background Leonhard Euler, 1736
From an actual map, then to a diagram, then to a graph, Leonhard Euler formulated a theorem, which started the study of Graph Theory. The theorem states that ”A connected graph G is Eulerian if and only if the degree of the vertex of G is even.”
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Convex Secure Domination in Graphs Under Some B
Denes Konig, 1936 It was only after 200 years that the first book on Graph Theory was published by Denes Konig. He wrote the book after having organized the works of other mathematicians and his own works. It was actively used in various areas, since then.
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Convex Secure Domination in Graphs Under Some B
Denes Konig, 1936 It was only after 200 years that the first book on Graph Theory was published by Denes Konig. He wrote the book after having organized the works of other mathematicians and his own works. It was actively used in various areas, since then. E.L. Enriquez & S.R. Canoy,Jr., 2015 In 2015, Dr. Enriquez and Dr. Canoy published an article entitled Secure Convex Domination in a Graph. This study motivated the researcher to explore a new domination parameter, the Convex Secure Domination in Graphs.
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Convex Secure Domination in Graphs Under Some B
Terms Used in the Study
Illustrations
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Convex Secure Domination in Graphs Under Some B
WORKING DEFINITIONS
Working Definitions
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Convex Secure Domination in Graphs Under Some B
WORKING DEFINITIONS
Working Definitions 1. Convex secure dominating set A convex dominating set S of G is a convex secure dominating set, if for each element u in V (G) \ S, there exists an element v in S such that uv ∈ E(G) and (S \ {v}) ∪ {u}, is a dominating set.
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Convex Secure Domination in Graphs Under Some B
WORKING DEFINITIONS
Working Definitions 1. Convex secure dominating set A convex dominating set S of G is a convex secure dominating set, if for each element u in V (G) \ S, there exists an element v in S such that uv ∈ E(G) and (S \ {v}) ∪ {u}, is a dominating set. 2. Convex secure domination number The convex secure domination number of G, denoted by γcs (G), is the minimum cardinality of a convex secure dominating set of G.
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Convex Secure Domination in Graphs Under Some B
WORKING DEFINITIONS
Working Definitions 1. Convex secure dominating set A convex dominating set S of G is a convex secure dominating set, if for each element u in V (G) \ S, there exists an element v in S such that uv ∈ E(G) and (S \ {v}) ∪ {u}, is a dominating set. 2. Convex secure domination number The convex secure domination number of G, denoted by γcs (G), is the minimum cardinality of a convex secure dominating set of G. 3. γcs -set A convex secure dominating set of cardinality γcs (G) will be called γcs (G)-set.
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Convex Secure Domination in Graphs Under Some B
RESULTS
Remark 2.1 A convex secure dominating set of a graph G is a convex dominating set and secure dominating set of G.
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Convex Secure Domination in Graphs Under Some B
RESULTS
Remark 2.1 A convex secure dominating set of a graph G is a convex dominating set and secure dominating set of G. Remark 2.2 Let G be a non-trivial connected graph. then
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Convex Secure Domination in Graphs Under Some B
RESULTS
Remark 2.1 A convex secure dominating set of a graph G is a convex dominating set and secure dominating set of G. Remark 2.2 Let G be a non-trivial connected graph. then (i) γ(G) ≤ γs (G) ≤ γcs (G); and
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Convex Secure Domination in Graphs Under Some B
RESULTS
Remark 2.1 A convex secure dominating set of a graph G is a convex dominating set and secure dominating set of G. Remark 2.2 Let G be a non-trivial connected graph. then (i) γ(G) ≤ γs (G) ≤ γcs (G); and (ii) 1 ≤ γcs (G) ≤ n.
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Convex Secure Domination in Graphs Under Some B
RESULTS
Theorem 2.3 Given positive integers k and n such that 1 ≤ k ≤ n, there exists a connected non-trivial graph G with |V (G)| = n and γcs (G) = k.
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Convex Secure Domination in Graphs Under Some B
RESULTS
Theorem 2.3 Given positive integers k and n such that 1 ≤ k ≤ n, there exists a connected non-trivial graph G with |V (G)| = n and γcs (G) = k. Corollary 2.4 The difference γcs (G) - γ(G) can be made arbitrarily large.
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Convex Secure Domination in Graphs Under Some B
RESULTS
Theorem 2.3 Given positive integers k and n such that 1 ≤ k ≤ n, there exists a connected non-trivial graph G with |V (G)| = n and γcs (G) = k. Corollary 2.4 The difference γcs (G) - γ(G) can be made arbitrarily large. Remark 2.5 Every clique dominating set is a convex dominating set.
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Convex Secure Domination in Graphs Under Some B
RESULTS for the Characterization of convex secure dominating sets with convex secure domination number of one
Theorem 2.6 Let G be a graph of order n ≥ 1. Then γcs (G) = 1 if and only if G is a complete graph.
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Convex Secure Domination in Graphs Under Some B
Characterization of convex secure dominating sets with convex secure domination number of two Theorem 2.7 Let G be a connected graph of order n ≥ 3. Then γcs (G) = 2 if and only if G is non-complete and there exists distinct and adjacent vertices x and y that dominate G and satisfy one of the following conditions:
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Convex Secure Domination in Graphs Under Some B
Characterization of convex secure dominating sets with convex secure domination number of two Theorem 2.7 Let G be a connected graph of order n ≥ 3. Then γcs (G) = 2 if and only if G is non-complete and there exists distinct and adjacent vertices x and y that dominate G and satisfy one of the following conditions: (i) N (x) \ {y} = N (y) \ {x} = V (G) \ {x, y}.
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Convex Secure Domination in Graphs Under Some B
Characterization of convex secure dominating sets with convex secure domination number of two Theorem 2.7 Let G be a connected graph of order n ≥ 3. Then γcs (G) = 2 if and only if G is non-complete and there exists distinct and adjacent vertices x and y that dominate G and satisfy one of the following conditions: (i) N (x) \ {y} = N (y) \ {x} = V (G) \ {x, y}. (ii) hN (x) \ N [y]i and hN (y) \ N [x]i are complete and for each u ∈ N (x) ∩ N (y) either h(N (x) \ N [y] ∪ {u}i or h(N (y) \ N [x]) ∪ {u}i is complete.
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Convex Secure Domination in Graphs Under Some B
Characterization of convex secure dominating sets with convex secure domination number of two Theorem 2.7 Let G be a connected graph of order n ≥ 3. Then γcs (G) = 2 if and only if G is non-complete and there exists distinct and adjacent vertices x and y that dominate G and satisfy one of the following conditions: (i) N (x) \ {y} = N (y) \ {x} = V (G) \ {x, y}. (ii) hN (x) \ N [y]i and hN (y) \ N [x]i are complete and for each u ∈ N (x) ∩ N (y) either h(N (x) \ N [y] ∪ {u}i or h(N (y) \ N [x]) ∪ {u}i is complete. (iii) N (x) \ {y} = V (G) \ {x, y}, N (x) \ N [y] 6= ∅ and hN (x) \ N [y]i is complete.
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Convex Secure Domination in Graphs Under Some B
RESULTS for a convex secure domination number of two
Theorem 2.7
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Convex Secure Domination in Graphs Under Some B
RESULTS for JOIN of Two Graphs
Theorem 2.8 Let G and H be connected non-complete graphs. Then a proper subset S of V (G + H) is a convex secure dominating set in G + H if and only if S is a clique secure dominating set in G + H.
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Convex Secure Domination in Graphs Under Some B
RESULTS for JOIN of Two Graphs Corollary 2.9 Let G and H be connected non-complete graphs. 2, if γcl (G) = 2 or γcl (H) = 2. 3, if SG of order m ≥ 2 and V (G) \ NG [SG ] are cliques in G or SH of order n ≥ 2 γcs (G + H) = and V (H) \ NH [SH ] are cliques in H, where SG ∈ V (G) and SH ∈ V (H). 4, if SG of order m ≥ 2 is clique in G and SH of order n ≥ 2 is clique in H.
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Convex Secure Domination in Graphs Under Some B
RESULTS for the Corona of Two Graphs
Remark 2.10 For any connected graph G, V (G) is a minimum dominating set in G ◦ H.
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Convex Secure Domination in Graphs Under Some B
RESULTS for the Corona of Two Graphs
Remark 2.10 For any connected graph G, V (G) is a minimum dominating set in G ◦ H. Theorem 2.11 Let G and H be non-trivial connected graphs. A non-empty subset S of V (G ◦ H) is a convex secure dominating set of G ◦ H if for each v ∈ V (G), one of the following conditions is satisfied.
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Convex Secure Domination in Graphs Under Some B
RESULTS for the Corona of Two Graphs
Remark 2.10 For any connected graph G, V (G) is a minimum dominating set in G ◦ H. Theorem 2.11 Let G and H be non-trivial connected graphs. A non-empty subset S of V (G ◦ H) is a convex secure dominating set of G ◦ H if for each v ∈ V (G), one of the following conditions is satisfied. (i) S = V (G) and H is complete.
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Convex Secure Domination in Graphs Under Some B
RESULTS for the Corona of Two Graphs
Remark 2.10 For any connected graph G, V (G) is a minimum dominating set in G ◦ H. Theorem 2.11 Let G and H be non-trivial connected graphs. A non-empty subset S of V (G ◦ H) is a convex secure dominating set of G ◦ H if for each v ∈ V (G), one of the following conditions is satisfied. (i) S = V (G) and H is complete. [ (ii) S = V (G) ∪ ( Sv ), and Sv is a convex set in H v . v∈V (G)
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Convex Secure Domination in Graphs Under Some B
RESULTS for the Corona of Two Graphs
Corollary 2.12 Let G and H be γcs (G ◦ H) =
connected graphs. |V (G)|, if H is a complete graph. 2|V (G)|, if γ(H) = 1 and H is a non-complete graph. (γ(H) + 1)|V (G)|, if γ(H) = |Sv | for each v ∈ V (G) and Sv ⊂ V (H v ).
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Convex Secure Domination in Graphs Under Some B
RESULTS
Remark 2.13 A clique secure dominating set S of a graph G is a clique and a secure dominating set of G.
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Convex Secure Domination in Graphs Under Some B
RESULTS
Remark 2.13 A clique secure dominating set S of a graph G is a clique and a secure dominating set of G. Remark 2.14 Every secure clique dominating set of a graph G is a clique secure dominating set of G.
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Convex Secure Domination in Graphs Under Some B
RESULTS for the Composition of Two Graphs
Theorem 2.15 Let G and H be connected non-complete [ graphs such that G has a clique dominating set. A subset C = [{x} × Tx ] where S ⊆ V (G) x∈S
and Tx ⊆ V (H) for each x ∈ S, is a convex secure dominating set of G[H] if and only if one of the following statements is satisfied:
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Convex Secure Domination in Graphs Under Some B
RESULTS for the Composition of Two Graphs
Theorem 2.15 Let G and H be connected non-complete [ graphs such that G has a clique dominating set. A subset C = [{x} × Tx ] where S ⊆ V (G) x∈S
and Tx ⊆ V (H) for each x ∈ S, is a convex secure dominating set of G[H] if and only if one of the following statements is satisfied: (i) S is a secure clique dominating set of G and |Tx | = 1 with diam(H) ≤ 4.
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Convex Secure Domination in Graphs Under Some B
RESULTS for the Composition of Two Graphs
Theorem 2.15 Let G and H be connected non-complete [ graphs such that G has a clique dominating set. A subset C = [{x} × Tx ] where S ⊆ V (G) x∈S
and Tx ⊆ V (H) for each x ∈ S, is a convex secure dominating set of G[H] if and only if one of the following statements is satisfied: (i) S is a secure clique dominating set of G and |Tx | = 1 with diam(H) ≤ 4. (ii) S is a clique secure dominating set of G and hTx i is a clique in H with |Tx | ≥ 2 for all x ∈ S where Tx is a clique secure dominating set of H whenever S = {x} is a dominating set of G.
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Convex Secure Domination in Graphs Under Some B
RESULTS for the Composition of Two Graphs Corollary 2.16 Let G and H be connected non-complete graphs with γ(G) = 1. Then if γ(H) = 1 and γcls (H) = 2, 2, γcs (G[H]) = or γcl (H) = 2 γcl (H), if H has a clique dominating set.
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Convex Secure Domination in Graphs Under Some B
RESULTS for the Cartesian Product of Two Graphs
Theorem 2.17 Let G[ and H be non-complete connected graphs. A non-empty subset C= ({x} × Tx ) is convex secure dominating set in GH if and x∈S
only if one of the following statements is satisfied.
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Convex Secure Domination in Graphs Under Some B
RESULTS for the Cartesian Product of Two Graphs
Theorem 2.17 Let G[ and H be non-complete connected graphs. A non-empty subset C= ({x} × Tx ) is convex secure dominating set in GH if and x∈S
only if one of the following statements is satisfied. (i) S is a convex secure dominating set in G and Tx = V (H) for each x ∈ S.
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Convex Secure Domination in Graphs Under Some B
RESULTS for the Cartesian Product of Two Graphs
Theorem 2.17 Let G[ and H be non-complete connected graphs. A non-empty subset C= ({x} × Tx ) is convex secure dominating set in GH if and x∈S
only if one of the following statements is satisfied. (i) S is a convex secure dominating set in G and Tx = V (H) for each x ∈ S. (ii) S = V (G) and Tx is a convex secure dominating set of H for each x ∈ S.
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Convex Secure Domination in Graphs Under Some B
RESULTS for the Cartesian Product of Two Graphs
Corollary 2.18 Let G and H be non-complete connected graphs. Then γcs (GH) = min{|V (G)|γcs (H), γcs (G)|V (H)|}.
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Convex Secure Domination in Graphs Under Some B
BIBLIOGRAPHY BOOKS Moise, E.E., & Downs, F.L.,Jr. (1975). Geometry. Addison-Wesley Secondary Mathematics Series, Canada, USA. eBOOKS Bondy, J.A., & Murty, U.S.R. (1982). Graph Theory with applications. Elsevier Science Publishing Co., Inc., New York. Harary, F. (1969). Graph Theory. Addison-Wesley Publishing Company, Inc., Canada, USA. Wilson, R.J. (1998). Introduction to Graph Theory (4th ed.). Addison-Wesley Longman Limited, England.
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Convex Secure Domination in Graphs Under Some B
PUBLISHED ARTICLES Canoy, S.R.,Jr., & Eballe, R.G.(2014). M-convex hulls in graphs resulting from some binary operations. Applied Mathematical Sciences, 8(88), 4389-4396. Castillano, E.C., Ugbinada, R.L., & Canoy, S.R.,Jr.(2014). Secure domination in the join of graphs. Applied Mathematical Sciences, 8(105), 5203-5211. Chang, G.T., Tong, L., & Wang, H., (2004). Geodetic spectra of graphs, European Journal of Combinatorics, 25, 383-391. Daniel, T.V., & Canoy, S.R.,Jr.(2015). Clique domination in a graph. Applied Mathematical Sciences, 9, 5749-5755. Desormeaux, W.J., Haynes, T.W., & Henning, M.A.(2014). Improved bounds on the domination number of a tree. Discrete Applied Mathematics, 177, 88-94. Enriquez, E.L., & Canoy, S.R.,Jr.(2015). Secure domination in a graph. International Journal of Mathematical Analysis, 9(7), 317-325. E.S.-E. and E.L.E.
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Farber, M., & Jamison, R.E.,Jr.(1987). On local convexity in graphs. Discrete Mathematics, North-Holland, 66, 231-247. Go, C.E., & Canoy, S.R.,Jr.(2011). Domination in the corona and join of graphs.International Mathematical Forum, 6(16), 763-771. Kiunisala, E.M., & Enriquez, E.L.(2016). On clique secure domination in graphs. Global Journal of Pure and Applied Mathematics, 12(3), 2075-2084. Klostermeyer, W.F., & Mynhardt, C.M.(2008). Secure domination and secure total domination in graphs. Discussiones Mathematicae, 28, 267-284.
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Labendia, M.A., & Canoy, S.R.,Jr.(2012). Convex domination in the composition and Cartesian product of graph. Czechoslovak Mathematical Journal, 62, 1003-1009. Leonida, R.E., & Canoy, S.R.,Jr.(2015). Weakly convexity and weakly convex domination in graphs. Applied Mathematical Sciences, 9(3), 137-141. Loquias, C.M., & Enriquez, E.L.(2016). On secure convex and restrained convex domination in graphs. International Journal of Applied Engineering Research ISSN 0973-4562, 11(7), 4707-4710. Nabiyeb, V., Cakirogla, U., Karak, H., Erumit, A.K., & Cabi, A. (2016). Applications of Graph Theory in an intelligent tutoring system for solving mathematical word problems. Eurasia Journal of Mathematics, Science & Technology Education, 12(4), 687-701.
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Convex Secure Domination in Graphs Under Some B
Raj, S.A.A., & Victor, S.P.,Jr.(2015). Algorithms to find geodetic numbers and edge geodetic numbers in graphs. Indian Journal of Science and Technology, 8(13). DOI:10.17485/ijst/2015 /v8i13/53160,July2015. Roushini, P., Pushpam, L., & Suseendran, C.(2015). Efficient secure domination in graphs. International Journal of Mathematics and Soft Computing,5(2),59-68. WEBSITE www.cisco.com/c/en/us/td/docs/voice ip comm/cucm/srnd/collab09 /clb09/netstruc.html Google map
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Convex Secure Domination in Graphs Under Some B
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Convex Secure Domination in Graphs Under Some B
