Secure Connected Domination in a Graph
|
|
- Ferdinand Harrington
- 5 years ago
- Views:
Transcription
1 International Journal of Mathematical Analysis Vol. 8, 2014, no. 42, HIKARI Ltd, Secure Connected Domination in a Graph Amerkhan G. Cabaro Department of Mathematics College of Natural Sciences and Mathematics Mindanao State University Marawi City, Philippines Sergio S. Canoy, Jr., and Imelda S. Aniversario Department of Mathematics & Statistics College of Sciences and Mathematics Mindanao State University- Iligan Institute of Technology 9200 Iligan City, Philippines Copyright c 2014 Amerkhan G. Cabaro, Sergio S. Canoy, Jr. and Imelda S. Aniversario. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Let G = (V (G), E(G)) be a connected simple graph. A connected dominating set S of V (G) is a secure connected dominating set of G if for each u V (G) \ S, there exists v S such that uv E(G) and the set (S \ {v} {u}) is a connected dominating set of G. The minimum cardinality of a secure connected dominating set of G, denoted by γ sc (G), is called the secure connected domination number of G. We characterized secure connected dominating set in terms of the concept of external private neighborhood of a vertex. Also, we give necessary and sufficient conditions for connected graphs to have secure connected domination number equal to 1 or 2. The secure connected domination numbers of graphs resulting from some binary operations are also investigated. Mathematics Subject Classification: 05C69
2 2066 Amerkhan G. Cabaro, Sergio S. Canoy, Jr. and Imelda S. Aniversario Keywords: dominating set, connected dominating set, secure connected dominating set, external private neighbors 1 Introduction Let G = (V (G), E(G)) be a simple graph with vertex set V (G) of finite order and edge set E(G). The neighborhood of v is the set N G (v) = N(v) = {u V (G) : uv E(G)}. If X V (G), then the open neighborhood of X is the set N G (X) = N(X) = {N G (v) : v X}. The closed neighborhood of X is N G [X] = N[X] = X N(X). A vertex w V (G) \ X is an X-external private neighbor of v X if N G (w) X = {v}. The set of all external private neighbors of v X is denoted by epn(v, X). A subset S of V (G) is a dominating set(ds) in G if for every u V (G)\S, there exists v S such that uv E(G), i.e., N[S] = V (G). The domination number of G is the minimum cardinality of a dominating set in G and is denoted by γ(g). A set S is said to be a secure dominating set in G if for every u V (G) \ S there exists v S such that uv E(G) and (S \ {v}) {u} is a dominating set. The minimum cardinality of a secure dominating set in G is called the secure domination number of G and is denoted by γ s (G). This variant of domination was introduced in [5] and several studies is found in [1, 2, 4] and elsewhere. Another variant of domination was introduced by Sampathkumar and Walikar in [8] which they called connected domination. A dominating set S is said to be connected dominating set(cds), if the induced subgraph S is connected. Let S be a connected dominating set in G. A vertex v S is said to S-defend u, where u V (G) \ S, if uv E(G) and (S \ {v}) {u} is a connected dominating set in G. S is a secure connected dominating set (SCDS) in G if for each u V (G) \ S, there exists v S such that v S-defends u. The secure connected domination number γ sc (G) of G is the smallest cardinality of a secure connected dominating set in G. A dominating set ( resp. secure connected dominating set ) S in G with S = γ(g) ( resp. S = γ sc (G) ) is called a γ-set ( resp. γ sc -set). In this paper, we initiate a study of secure connected domination in graphs. Any undefined terms maybe found in [3] or [6]. 2 Results and Discussion Remark 2.1 Let G be a connected graph of order n. Then 1 γ sc (G) n. Theorem 2.2 Let S be a connected dominating set of G, v S and u V (G) \ S with uv E(G). v S-defends u if and only if epn(v, S) N G [u] and V (C) N G (u), for every component C of S \ {v}.
3 Secure connected domination in a graph 2067 Proof : Suppose v S-defends u. Let w epn(v, S). Then w / S and N G (w) S = {v}. If w = u, then w N G [u]. If w u, then w / T, where T = (S \ {v}) {u}. Since T is a dominating set in G and N G (w) S = {v}, w N G (u). Hence, epn(v, S) N G [u]. Suppose that V (C) N G (u) = for some component C of S \ {v}. Then there is no path joining any a V (C) and u; hence T is not connected which is contrary to the assumption that v S-defends u. Now we suppose that epn(v, S) N G [u] and V (C) N G (u), for every component C of S \ {v}. Let w V (G) \ T, where T = (S \ {v}) {u}. If w = v, then uw E(G). If w v, then w / S. If w epn(v, S), then uw E(G). If w / epn(v, S), then N G (w) S {v}. Hence, there exists t T \ {v} T such that tw E(G). Thus, T is a dominating set in G. It remains to show that T is connected. Let a, b T, where a b and ab / E(G). Consider the following cases: Case 1: a = u and b V (D 1 ), where D 1 is a component of S \ {v} Let z V (D 1 ) N G (a). Then az E(G) and z V (D 1 ). Since D 1 is connected, there is a path, say P (z, b), joining z and b. Now, the path P (z, b) together with the edge az forms a path P (a, b) joining a and b. Case 2: a and b are in the same component D of S \ {v} Since D is connected, there is a path joining a and b. Case 3: a V (D 1 ) and b V (D 2 ), where D 1 and D 2 are different components of S \ {v} Let w V (D 1 ) N G (u) and x V (D 2 ) N G (u). Since D 1 is connected, there is a path P (a, w) joining a and w. Similarly, there is also a path P (x, b) joining x and b. The paths P (a, w) and P (x, b), together with the edges uw and ux form a path P (a, b) from a to b. Thus, in either case, T is connected. Corollary 2.3 Let S be a connected dominating set in G. Then S is a secure connected dominating set in G if and only if for every u V (G) \ S, v S N G (u) such that (i) epn(v, S) N G [u], and (ii) V (C) N G (u), for every component C of S \ {v}. Proof : Suppose S is an SCDS in G. Then for every u V (G) \ S, there exists v S N G (u) such that (S \ {v}) {u} is a CDS in G. This means that v S defends u. By Theorem 2.2, (i) and (ii) hold. The converse follows immediately. A leaf u of a graph G is a vertex of degree one and the support vertex of a leaf u is the unique vertex v such that uv E(G). We denote by L(G) and S(G) be the set of leaves and support vertices of G, respectively.
4 2068 Amerkhan G. Cabaro, Sergio S. Canoy, Jr. and Imelda S. Aniversario Theorem 2.4 Let G be a connected graph of order n 3 and let X be a secure connected dominating set of G. Then (i) L(G) X and S(G) X (ii) No vertex in L(G) S(G) is an X-defender. Proof : Suppose that x L(G). Let y S(G) such that xy E(G). Let S = (X \ {y}) {x}. Since S is not connected, x X. Suppose x S(G). Let z L(G) N G (x). Then z X. Since X is connected, x X. Let v L(G) S(G) and u V (G) \ X. Now v L(G) S(G) = v L(G) or v S(G). If v L(G), then uv / E(G). If v S(G), ( then ) there exists z V (G) such that N(z) = {v}. By (i), z X. Thus X \ {v} {u} is not connected. Thus, in either case, v does not X-defend u. Theorem 2.5 Let n be a positive integer. Then (i) γ sc (K n ) = 1 for all n 2 { 1, if n = 2 (ii) γ sc (P n ) = n, if n 3 { 1, if n = 3 (iii) γ sc (C n ) = n 1, if n > 3 Proof : First suppose that G = K n. Then S = {v} V (G) is a connected dominating set of G. Now x V (G), (S \ {v}) {x} = {x} which is a CDS of G. Thus S = {v} is an SCDS of G. Therefore, γ sc (G) 1. By Remark 2.1, it follows that γ sc (G) = 1. Next, let P n be a path of order n 2. For n = 2, P 2 = K2 and γ sc (P 2 ) = 1. Suppose n 3. Let S be a γ sc -set of P n = [v 1, v 2,..., v n ]. Then v 1, v n S by Theorem 2.4. Since S is connected, S = V (P n ). Thus γ sc (P n ) = S = n. Finally, let C n be a cycle of order n 3. Suppose n = 3, then C 3 = K3 and γ sc (C 3 ) = 1. For n > 3, let C n = [v 1, v 2,..., v n, v 1 ] and let S = {v 1, v 2,..., v n 1 }. Clearly, S is an SCDS in C n. Let S be a γ sc -set of C n. Since S is an SCDS in C n, S S = n 1. Suppose S = k < n 1. Since S is connected, V (C n ) \ S is connected, where V (C n ) \ S = m 2. Moreover, since S is a dominating set, m = 2. Let V (C n ) \ S = {v i, v i+1 }, i = 1, 2,..., n 1. Note that (S \ {v i 1 }) {v i } = {v 1, v 2,..., v i 2, v i, v i+1,..., v n } is not connected, which is contrary to the assumption that S = k < n 1. This shows that S = k = n 1. Consequently, γ sc (C n ) = n 1.
5 Secure connected domination in a graph 2069 Theorem 2.6 Let G be a connected graph of order n. Then γ sc (G) = 1 if and only if G = K n. Proof : Suppose γ sc (G) = 1 and let S = {v} be an SCDS of G. Suppose G K n, then there exists x, y V (K n ) such that d(x, y) = 2. Then (S \ {v}) {x} = {x}, which is not a dominating set of G, since xy / E(K n ). Therefore, G = K n. The converse is true by Theorem 2.5. The following Theorem shows that it is not possible to find a non-complete connected graph G such that γ(g) = γ sc (G). Theorem 2.7 Let G be a non-complete connected graph and let S be a secure connected dominating set in G. Then the set S \ {v} is a dominating set for every v S. In particular, 1 + γ(g) γ sc (G). Proof : Let v S and T = S \ {v}. We will show that T is a dominating set in G. Let w V (G) \ T. If w = v, then u S such that uw E(G) because S is connected. If w v, then w / S. Since S is a secure dominating set in G, there exists x S N G (w) such that x S defends w. If x v, then x T. If x = v, then (S \ {v}) {w} is a CDS. This implies that u T such that uw E(G). Therefore, in any case, T = S \ {v} is a DS. Thus, 1 + γ(g) S. If, in particular, S is a γ sc -set, then 1 + γ(g) γ sc (G). Theorem 2.8 Let G be a connected graph of order n 4. Then γ sc (G) = 2 if and only if there exists a non-complete graph H such that G = K 2 + H. Proof : Suppose G = K 2 + H, for some non-complete graph H. Then G is not complete and by Theorem 2.6, it follows that γ sc (G) 2. Let S = {x, y}, where x, y V (K 2 ). Then S is a connected dominating set in G. Let z V (G)\S. Then z V (H), xz E(G) and (S \ {x}) {z} = {y, z}a connected dominating set of G. Thus S is an SCDS in G. Therefore, γ sc (G) = 2. For the converse, suppose that γ sc (G) = 2. Let S = {x, y} be a γ sc -set of G. Let z V (G) \ S. Suppose xz / E(G). Since S is an SCDS of G, yz E(G) and S = {x, z} is a connected dominating set of G. This is not possible because xz / E(G). Therefore, xz E(G). Similarly, yz E(G). Let H = V (G) \ S and let K 2 = S. Then G = K 2 + H. Moreover, since G is not complete, H is non-complete. Corollary 2.9 Let G be a non-complete connected graph and n 2. Then γ sc (G + K n ) = 2. Corollary 2.10 Let G be a non-complete connected graph. Then γ sc (K 1 + G) = 2 if and only if one of the following is true: (i) γ(g) = 1
6 2070 Amerkhan G. Cabaro, Sergio S. Canoy, Jr. and Imelda S. Aniversario (ii) γ sc (G) = 2 Proof : Suppose γ sc (K 1 + G) = 2 and let V (K 1 ) = {a}. Let S = {x, b} be an SCDS in K 1 + G, where b V (G). Consider the following cases: Case 1. Suppose x = a. If {b} were not a dominating set of G, then y V (G) \ {b} such that by / E(G). Since S is an SCDS of K 1 + G, S \ {x} {y} = {y, b} is a connected dominating set of K 1 + G. This, however, s impossible to happen since by / E(G). Thus, {b} is a dominating set of G. This shows that γ(g) = 1. Case 2. Suppose x a. Then x G. Hence, S is an SCDS in G. Since K 1 + G is not complete, it follows that γ sc (G) = S = 2. For the converse, suppose first that γ(g) = 1, say {b} is a dominating set of G. Let H = V (G) \ {b}. Then H is non-complete and K 1 +G {a, b} +H. By Theorem 2.8, γ sc (K 1 + G) = 2. Suppose now that γ sc (G) = 2. Then by Theorem 2.8, G = K 2 + H, where H is a non-complete graph. Let H = a + H. Then H is non-complete and K 1 + G K 2 + H. Therefore, by Theorem 2.8, γ sc (K 1 + G) = 2. Theorem 2.11 Let G be a non-complete connected graph and K 1 = v. Then S V (K 1 + G) is a SCDS of K 1 + G iff one of the following holds: (i) S V (G) and S is a SCDS of G. (ii) v S and S \ {v} is a dominating set in G. Proof : Suppose S V (K 1 + G) is an SCDS in K 1 + G. If S V (G), then S is an SCDS of G. Suppose v S and let x V (G) \ (S \ {v}). Suppose xy E(K 1 + G) for all y V (G) \ (S \ {v}). Then (S \ {v}) {x} is not connected, contrary to our assumption that S is an SCDS in K 1 + G. Thus, S \ {v} is a dominating set in G. For the converse suppose first that S is an SCDS in G. Then S is an SCDS in K 1 + G. Next suppose that v S and S \ {v} is a dominating set in G. Then S is a connected dominating set in K 1 + G. Let z V (K 1 + G) \ S. Then z V (G) \ (S \ {v}). Since S \ {v} is a DS in G, there exists x S \ {v} such that xz E(G) and (S \ {x}) {z} is a CDS in K 1 + G. Thus, S is an SCDS in K 1 + G. Corollary 2.12 Let G be a non-complete connected graph. Then γ sc (K 1 + G) = min{γ sc (G), γ(g) + 1}.
7 Secure connected domination in a graph 2071 Proof : Let S 1 and S 2 be, respectively, γ sc -set and γ-set of G. By Theorem 2.11, S 1 and S 2 V (K 1 ) are SCDS in K 1 + G. Thus, γ sc (K 1 + G) min{ S 1, S 2 V (K 1 ) } = min{γ sc (G), γ(g) + 1}. Let r = min{γ sc (G), γ(g) + 1}. Let S be a γ sc -set of K 1 + G. By Theorem 2.11, γ sc (K 1 + G) = S r. This proves the desired equality. From this result, we can readily solve for the secure connected domination number of the graphs: F n and W n. Remark 2.13 Let F n, and W n be the fan and wheel of order n + 1, respectively. Then { 1, if n = 1 (i) γ sc (F n ) = n , if n 2 (ii) γ sc (W n ) = { 1, if n = 3 n 3 + 1, if n 4. Remark 2.14 Corollary 2.10 also follows from Corollary Let G and H be non-complete connected graphs. Then G + H is noncomplete. Thus γ sc (G + H) 2. Now, choose a, b V (G) and x, y V (H). Then S = {a, b, x, y} is a SCDS in G + H. It follows that γ sc (G + H) 4. This result is stated formally in the following remark. Remark 2.15 Let G and H be non-complete connected graphs. γ sc (G + H) 4. Then 2 Theorem 2.16 Let G and H be non-complete connected graphs. Then γ sc (G+ H) = 2 if and only if one of the following holds: (i) γ sc (G) = 2. (ii) γ sc (H) = 2. (iii) γ(g) = 1 and γ(h) = 1. Proof : Suppose γ sc (G + H) = 2. Then by Theorem 2.8, G + H = K 2 + H 1, where H 1 is a non-complete graph. Let V (K 2 ) = {x, y} = S, then S is a SCDS in G + H. Consider the following cases: Case 1: Suppose x, y V (G) Then G = K 2 + J, where J is a non-complete subgraph of G. Hence by Theorem 2.8, γ sc (G) = 2. Case 2: Suppose x, y V (H) As in case 1, γ sc (H) = 2.
8 2072 Amerkhan G. Cabaro, Sergio S. Canoy, Jr. and Imelda S. Aniversario Case 3: Suppose x V (G) and y V (H) Let z V (G) \ {x}. Since G + H = K 2 + H 1, xz E(G). Hence {x} is a dominating set in G. It follows that γ(g) = 1. Similarly, γ(h) = 1. For the converse, suppose that γ sc (G) = 2. Then by Theorem 2.8, G = K 2 + J, where J is a non-complete graph. Thus, G + H = (K 2 + J) + H = K 2 + H 1, where H 1 = J + H is non-complete. It follows from Theorem 2.8 that γ sc (G + H) = 2. Similarly, γ sc (G + H) = 2 if γ sc (H) = 2 holds. Suppose (iii) holds. Let {x} and {y} be a dominating set of G and H, respectively. Then G = x + J and H = y + K where J and K are non-complete graphs. Then, G+H = ( x +J ) + ( y +K ) = K 2 +H 1, where V (K 2 ) = {x, y} and H 1 = J + K is a non-complete graph. Hence by Theorem 2.8, γ sc (G + H) = 2. Theorem 2.17 Let G and H be non-complete connected graph such that γ sc (G+ H) 2. Then γ sc (G + H) = 3 if and only if one of the following holds: (i) γ sc (G) = 3 or γ sc (H) = 3 (ii) γ(g) = 2 or γ(h) = 2. Proof : Suppose that γ sc (G + H) = 3. Let S = {x, y, z} be a SCDS in G + H. Consider the following cases: Case 1: Suppose S V (G) Then S is an SCDS in G. It follows that γ sc (G) = 3 Case 2: If S H Then S is an SCDS in H and γ sc (H) = 3. Case 3: Suppose V (G) S and V (H) S Let w( V (G) )\ {x, y}. Suppose that w / N G (X). Then wz E(G + H). Hence, S \ {z} {w} = {x, y, w} is not a CDS, which is contrary to our assumption about S. Therefore, N G [X] = V (G), that is, X is a DS in G. Since G is non-complete, γ(g) = 2. For the converse, suppose (ii) holds, say γ(g) = 2. Let X = {a, b} be a DS in G. Pick c V (H) and set S = {a, b, c}. Let z V (G + H) \ S. Then z V (G) or z V (H). If z V (G), then z N G (X), say az E(G + H), and S = {b, z, c} is a CDS in G + H. If z V (H), then az E(G + H) and S = {b, z, c} is a CDS in G + H. Thus, S is an SCDS in G + H. Therefore γ sc (G + H) = 3. We obtain the same conclusion if we assume that γ(h) = 2. Suppose (i) holds, say γ sc (H) = 3. Let S = {a, b, c} be a SCDS in H. Then, clearly S is a SCDS in G + H. Therefore, γ sc (G + H) = 3. Theorem 2.18 Let G be a non-trivial connected graph and let H be any graph. Then C V (G H) is a secure connected dominating set in G H if and only if C = V (G) ( v V (G) S v ), where each S v is a dominating set in H v.
9 Secure connected domination in a graph 2073 Proof : Suppose that C V (G H) is a secure connected dominating set in G H and let v V (G). Choose w V (G) \ {v}. Since C is a DS, V (v + H v ) C and V (w + H w ) C. Since C is connected, v C. Let S v = V (H v ) C and x V (H v ) \ S v. Since C is an SCDS of G H, there exists y C such that xy E(G H) and (C \ {y}) {x} is a CDS. Note that since (C \ {v}) {x} is not connected in G H, it follows that y v. Thus, y S v, showing that S v is a DS in H v. Conversely, suppose that C = V (G) ( v V (G) S v ). Then, clearly, C is a CDS in G H. Let x V (G H)\C and let v V (G) such that x V (H v )\S v. Since S v is a DS in H v, there exists z S v such that xz E(G H). Also, (C \ {x}) {z} is a CDS in G H. Therefore, C is an SCDS in G H. Corollary 2.19 Let G be a non-trivial connected graph of order m and let H be any graph of order m. Then γ sc (G H) = m(1 + γ(h)). Proof : Let S be a γ-set of H. For each v V (H), choose S v V (H v ) such that S v = S. Then, by Theorem 2.18, C = V (G) ( Sv ) is an SCDS in G H. Hence, γ sc (G H) m + S v = m + mγ(h) = m(1 + γ(h)). On the other hand, if C is a γ sc -set in G H, then C = V (G) ( Sv), where each Sv is a DS in H v, by Theorem Thus, γ sc (G H) = C = m + Sv m + mγ(h) = m(1 + γ(h)). This proves the desired equality. Acknowledgements. The researcher would like to thank the Commission of Higher Education of the Republic of the Philippines for the partial financial support extended through its Faculty Development Program Phase II. References [1] A.P. Burger, M.A. Henning, and J. H. Van Vuuren, Vertex Cover and Secure Domination in Graph,Quaestiones Mathematicae, 31:2 (2008), [2] E. Castillano, R.A. Ugbinada and S. Canoy, Jr., Secure Domination in the Join of Graphs, Applied Mathematical Sciences, submitted. [3] G. Chartrand and L. Lesniak, Graphs and Digraphs, Third Edition, Chapman and Hall, [4] E.J. Cockayne, O. Favaron, and C.M. Mynhardt, Secure Domination, Weak Roman Domination and Forbidden Subgraphs, Bull. Inst. Combin. Appl., 39 (2003), [5] E.J. Cockayne, P.J.P. Grobler, W.R. Grundlingh, J. Munganga, and J.H. Van Vuuren, Protection of a Graph, Util. Math., 67 (2005),
10 2074 Amerkhan G. Cabaro, Sergio S. Canoy, Jr. and Imelda S. Aniversario [6] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater(eds), Fundamentals of Domination in Graphs, Marcel Dekker, Inc. New York, [7] S.T. Hedetniemi, R.C. and Laskar, Connected Domination in Graphs, In B. Bollobás, editor, Graph Theory and Combinatorics, Academic Press, London, [8] E. Sampathkumar, and H.B. Walikar, The Connected Domination of a Graph, Math. Phys. Sci. 13 (1979), Received: July 1, 2014
p-liar s Domination in a Graph
Applied Mathematical Sciences, Vol 9, 015, no 107, 5331-5341 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/101988/ams0155749 p-liar s Domination in a Graph Carlito B Balandra 1 Department of Arts and Sciences
More informationRestrained Weakly Connected Independent Domination in the Corona and Composition of Graphs
Applied Mathematical Sciences, Vol. 9, 2015, no. 20, 973-978 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121046 Restrained Weakly Connected Independent Domination in the Corona and
More informationSecure Weakly Connected Domination in the Join of Graphs
International Journal of Mathematical Analysis Vol. 9, 2015, no. 14, 697-702 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.519 Secure Weakly Connected Domination in the Join of Graphs
More informationSecure Weakly Convex Domination in Graphs
Applied Mathematical Sciences, Vol 9, 2015, no 3, 143-147 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ams2015411992 Secure Weakly Convex Domination in Graphs Rene E Leonida Mathematics Department
More informationAnother Look at p-liar s Domination in Graphs
International Journal of Mathematical Analysis Vol 10, 2016, no 5, 213-221 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ijma2016511283 Another Look at p-liar s Domination in Graphs Carlito B Balandra
More informationRestrained Independent 2-Domination in the Join and Corona of Graphs
Applied Mathematical Sciences, Vol. 11, 2017, no. 64, 3171-3176 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.711343 Restrained Independent 2-Domination in the Join and Corona of Graphs
More informationLocating-Dominating Sets in Graphs
Applied Mathematical Sciences, Vol. 8, 2014, no. 88, 4381-4388 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.46400 Locating-Dominating Sets in Graphs Sergio R. Canoy, Jr. 1, Gina A.
More informationOn Pairs of Disjoint Dominating Sets in a Graph
International Journal of Mathematical Analysis Vol 10, 2016, no 13, 623-637 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ijma20166343 On Pairs of Disjoint Dominating Sets in a Graph Edward M Kiunisala
More informationSecure Domination in Graphs
Int. J. Advance Soft Compu. Appl, Vol. 8, No. 2, July 2016 ISSN 2074-8523 Secure Domination in Graphs S.V. Divya Rashmi 1, S. Arumugam 2, and Ibrahim Venkat 3 1 Department of Mathematics Vidyavardhaka
More informationInduced Cycle Decomposition of Graphs
Applied Mathematical Sciences, Vol. 9, 2015, no. 84, 4165-4169 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.5269 Induced Cycle Decomposition of Graphs Rosalio G. Artes, Jr. Department
More informationInverse Closed Domination in Graphs
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 2 (2016), pp. 1845-1851 Research India Publications http://www.ripublication.com/gjpam.htm Inverse Closed Domination in
More informationApplied Mathematics Letters
Applied Mathematics Letters 23 (2010) 1295 1300 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml The Roman domatic number of a graph S.M.
More informationIndependent Transversal Equitable Domination in Graphs
International Mathematical Forum, Vol. 8, 2013, no. 15, 743-751 HIKARI Ltd, www.m-hikari.com Independent Transversal Equitable Domination in Graphs Dhananjaya Murthy B. V 1, G. Deepak 1 and N. D. Soner
More informationOn Disjoint Restrained Domination in Graphs 1
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 3 (2016), pp. 2385-2394 Research India Publications http://www.ripublication.com/gjpam.htm On Disjoint Restrained Domination
More informationMore on Tree Cover of Graphs
International Journal of Mathematical Analysis Vol. 9, 2015, no. 12, 575-579 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.410320 More on Tree Cover of Graphs Rosalio G. Artes, Jr.
More informationVertices contained in all or in no minimum k-dominating sets of a tree
AKCE Int. J. Graphs Comb., 11, No. 1 (2014), pp. 105-113 Vertices contained in all or in no minimum k-dominating sets of a tree Nacéra Meddah and Mostafa Blidia Department of Mathematics University of
More informationDouble Total Domination on Generalized Petersen Graphs 1
Applied Mathematical Sciences, Vol. 11, 2017, no. 19, 905-912 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.7114 Double Total Domination on Generalized Petersen Graphs 1 Chengye Zhao 2
More informationRoman dominating influence parameters
Roman dominating influence parameters Robert R. Rubalcaba a, Peter J. Slater b,a a Department of Mathematical Sciences, University of Alabama in Huntsville, AL 35899, USA b Department of Mathematical Sciences
More informationA Characterization of the Cactus Graphs with Equal Domination and Connected Domination Numbers
International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 7, 275-281 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2017.7932 A Characterization of the Cactus Graphs with
More informationINDEPENDENT TRANSVERSAL DOMINATION IN GRAPHS
Discussiones Mathematicae Graph Theory 32 (2012) 5 17 INDEPENDENT TRANSVERSAL DOMINATION IN GRAPHS Ismail Sahul Hamid Department of Mathematics The Madura College Madurai, India e-mail: sahulmat@yahoo.co.in
More informationNordhaus Gaddum Bounds for Independent Domination
Nordhaus Gaddum Bounds for Independent Domination Wayne Goddard 1 Department of Computer Science, University of Natal, Durban 4041, South Africa Michael A. Henning School of Mathematics, Statistics and
More information1-movable Restrained Domination in Graphs
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 6 (2016), pp. 5245-5225 Research India Publications http://www.ripublication.com/gjpam.htm 1-movable Restrained Domination
More informationNote on Strong Roman Domination in Graphs
Applied Mathematical Sciences, Vol. 12, 2018, no. 11, 55-541 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.851 Note on Strong Roman Domination in Graphs Jiaxue Xu and Zhiping Wang Department
More informationRoman domination perfect graphs
An. Şt. Univ. Ovidius Constanţa Vol. 19(3), 2011, 167 174 Roman domination perfect graphs Nader Jafari Rad, Lutz Volkmann Abstract A Roman dominating function on a graph G is a function f : V (G) {0, 1,
More informationRelations between edge removing and edge subdivision concerning domination number of a graph
arxiv:1409.7508v1 [math.co] 26 Sep 2014 Relations between edge removing and edge subdivision concerning domination number of a graph Magdalena Lemańska 1, Joaquín Tey 2, Rita Zuazua 3 1 Gdansk University
More informationLocating-Total Dominating Sets in Twin-Free Graphs: a Conjecture
Locating-Total Dominating Sets in Twin-Free Graphs: a Conjecture Florent Foucaud Michael A. Henning Department of Pure and Applied Mathematics University of Johannesburg Auckland Park, 2006, South Africa
More informationALL GRAPHS WITH PAIRED-DOMINATION NUMBER TWO LESS THAN THEIR ORDER. Włodzimierz Ulatowski
Opuscula Math. 33, no. 4 (2013), 763 783 http://dx.doi.org/10.7494/opmath.2013.33.4.763 Opuscula Mathematica ALL GRAPHS WITH PAIRED-DOMINATION NUMBER TWO LESS THAN THEIR ORDER Włodzimierz Ulatowski Communicated
More informationGLOBAL MINUS DOMINATION IN GRAPHS. Communicated by Manouchehr Zaker. 1. Introduction
Transactions on Combinatorics ISSN (print): 2251-8657, ISSN (on-line): 2251-8665 Vol. 3 No. 2 (2014), pp. 35-44. c 2014 University of Isfahan www.combinatorics.ir www.ui.ac.ir GLOBAL MINUS DOMINATION IN
More informationSEMI-STRONG SPLIT DOMINATION IN GRAPHS. Communicated by Mehdi Alaeiyan. 1. Introduction
Transactions on Combinatorics ISSN (print): 2251-8657, ISSN (on-line): 2251-8665 Vol. 3 No. 2 (2014), pp. 51-63. c 2014 University of Isfahan www.combinatorics.ir www.ui.ac.ir SEMI-STRONG SPLIT DOMINATION
More informationEdge Fixed Steiner Number of a Graph
International Journal of Mathematical Analysis Vol. 11, 2017, no. 16, 771-785 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2017.7694 Edge Fixed Steiner Number of a Graph M. Perumalsamy 1,
More informationLocating Chromatic Number of Banana Tree
International Mathematical Forum, Vol. 12, 2017, no. 1, 39-45 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.610138 Locating Chromatic Number of Banana Tree Asmiati Department of Mathematics
More informationAALBORG UNIVERSITY. Total domination in partitioned graphs. Allan Frendrup, Preben Dahl Vestergaard and Anders Yeo
AALBORG UNIVERSITY Total domination in partitioned graphs by Allan Frendrup, Preben Dahl Vestergaard and Anders Yeo R-2007-08 February 2007 Department of Mathematical Sciences Aalborg University Fredrik
More information2-bondage in graphs. Marcin Krzywkowski*
International Journal of Computer Mathematics Vol. 00, No. 00, January 2012, 1 8 2-bondage in graphs Marcin Krzywkowski* e-mail: marcin.krzywkowski@gmail.com Department of Algorithms and System Modelling
More informationLower bounds on the minus domination and k-subdomination numbers
Theoretical Computer Science 96 (003) 89 98 www.elsevier.com/locate/tcs Lower bounds on the minus domination and k-subdomination numbers Liying Kang a;, Hong Qiao b, Erfang Shan a, Dingzhu Du c a Department
More informationDomination in Cayley Digraphs of Right and Left Groups
Communications in Mathematics and Applications Vol. 8, No. 3, pp. 271 287, 2017 ISSN 0975-8607 (online); 0976-5905 (print) Published by RGN Publications http://www.rgnpublications.com Domination in Cayley
More informationA characterization of diameter-2-critical graphs with no antihole of length four
Cent. Eur. J. Math. 10(3) 2012 1125-1132 DOI: 10.2478/s11533-012-0022-x Central European Journal of Mathematics A characterization of diameter-2-critical graphs with no antihole of length four Research
More informationExtremal Problems for Roman Domination
Extremal Problems for Roman Domination Erin W Chambers, Bill Kinnersley, Noah Prince, Douglas B West Abstract A Roman dominating function of a graph G is a labeling f : V (G) {0, 1, 2} such that every
More informationThe domination game played on unions of graphs
The domination game played on unions of graphs Paul Dorbec 1,2 Gašper Košmrlj 3 Gabriel Renault 1,2 1 Univ. Bordeaux, LaBRI, UMR5800, F-33405 Talence 2 CNRS, LaBRI, UMR5800, F-33405 Talence Email: dorbec@labri.fr,
More informationA note on the total domination number of a tree
A note on the total domination number of a tree 1 Mustapha Chellali and 2 Teresa W. Haynes 1 Department of Mathematics, University of Blida. B.P. 270, Blida, Algeria. E-mail: m_chellali@yahoo.com 2 Department
More informationDouble Total Domination in Circulant Graphs 1
Applied Mathematical Sciences, Vol. 12, 2018, no. 32, 1623-1633 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.811172 Double Total Domination in Circulant Graphs 1 Qin Zhang and Chengye
More informationGeneralized connected domination in graphs
Discrete Mathematics and Theoretical Computer Science DMTCS vol. 8, 006, 57 64 Generalized connected domination in graphs Mekkia Kouider 1 and Preben Dahl Vestergaard 1 Laboratoire de Recherche en Informatique,
More informationGENERALIZED INDEPENDENCE IN GRAPHS HAVING CUT-VERTICES
GENERALIZED INDEPENDENCE IN GRAPHS HAVING CUT-VERTICES Vladimir D. Samodivkin 7th January 2008 (Dedicated to Mihail Konstantinov on his 60th birthday) Abstract For a graphical property P and a graph G,
More informationThe Rainbow Connection of Windmill and Corona Graph
Applied Mathematical Sciences, Vol. 8, 2014, no. 128, 6367-6372 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.48632 The Rainbow Connection of Windmill and Corona Graph Yixiao Liu Department
More informationEXACT DOUBLE DOMINATION IN GRAPHS
Discussiones Mathematicae Graph Theory 25 (2005 ) 291 302 EXACT DOUBLE DOMINATION IN GRAPHS Mustapha Chellali Department of Mathematics, University of Blida B.P. 270, Blida, Algeria e-mail: mchellali@hotmail.com
More informationProperties of independent Roman domination in graphs
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 5 (01), Pages 11 18 Properties of independent Roman domination in graphs M. Adabi E. Ebrahimi Targhi N. Jafari Rad M. Saied Moradi Department of Mathematics
More informationSome Properties of D-sets of a Group 1
International Mathematical Forum, Vol. 9, 2014, no. 21, 1035-1040 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.45104 Some Properties of D-sets of a Group 1 Joris N. Buloron, Cristopher
More informationOn Dominator Colorings in Graphs
On Dominator Colorings in Graphs Ralucca Michelle Gera Department of Applied Mathematics Naval Postgraduate School Monterey, CA 994, USA ABSTRACT Given a graph G, the dominator coloring problem seeks a
More informationAxioms of Countability in Generalized Topological Spaces
International Mathematical Forum, Vol. 8, 2013, no. 31, 1523-1530 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2013.37142 Axioms of Countability in Generalized Topological Spaces John Benedict
More information1-movable Independent Outer-connected Domination in Graphs
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 1 (2017), pp. 41 49 Research India Publications http://www.ripublication.com/gjpam.htm 1-movable Independent Outer-connected
More informationDomination and Total Domination Contraction Numbers of Graphs
Domination and Total Domination Contraction Numbers of Graphs Jia Huang Jun-Ming Xu Department of Mathematics University of Science and Technology of China Hefei, Anhui, 230026, China Abstract In this
More informationk-tuple Domatic In Graphs
CJMS. 2(2)(2013), 105-112 Caspian Journal of Mathematical Sciences (CJMS) University of Mazandaran, Iran http://cjms.journals.umz.ac.ir ISSN: 1735-0611 k-tuple Domatic In Graphs Adel P. Kazemi 1 1 Department
More informationD.K.M.College for Women (Autonomous), Vellore , Tamilnadu, India.
American International Journal of Research in Science, Technology, Engineering & Mathematics Available online at http://www.iasir.net ISSN (Print): 38-3491, ISSN (Online): 38-3580, ISSN (CD-ROM): 38-369
More informationFundamental Dominations in Graphs
Fundamental Dominations in Graphs arxiv:0808.4022v1 [math.co] 29 Aug 2008 Arash Behzad University of California, LosAngeles abehzad@ee.ucla.edu Mehdi Behzad Shahid Beheshti University, Iran mbehzad@sharif.edu
More informationNORDHAUS-GADDUM RESULTS FOR WEAKLY CONVEX DOMINATION NUMBER OF A GRAPH
Discussiones Mathematicae Graph Theory 30 (2010 ) 257 263 NORDHAUS-GADDUM RESULTS FOR WEAKLY CONVEX DOMINATION NUMBER OF A GRAPH Magdalena Lemańska Department of Applied Physics and Mathematics Gdańsk
More informationH Paths in 2 Colored Tournaments
International Journal of Contemporary Mathematical Sciences Vol. 10, 2015, no. 5, 185-195 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2015.5418 H Paths in 2 Colo Tournaments Alejandro
More informationSTRUCTURE OF THE SET OF ALL MINIMAL TOTAL DOMINATING FUNCTIONS OF SOME CLASSES OF GRAPHS
Discussiones Mathematicae Graph Theory 30 (2010 ) 407 423 STRUCTURE OF THE SET OF ALL MINIMAL TOTAL DOMINATING FUNCTIONS OF SOME CLASSES OF GRAPHS K. Reji Kumar Department of Mathematics N.S.S College,
More informationIrredundance saturation number of a graph
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 46 (2010), Pages 37 49 Irredundance saturation number of a graph S. Arumugam Core Group Research Facility (CGRF) National Center for Advanced Research in Discrete
More informationON DOMINATING THE CARTESIAN PRODUCT OF A GRAPH AND K 2. Bert L. Hartnell
Discussiones Mathematicae Graph Theory 24 (2004 ) 389 402 ON DOMINATING THE CARTESIAN PRODUCT OF A GRAPH AND K 2 Bert L. Hartnell Saint Mary s University Halifax, Nova Scotia, Canada B3H 3C3 e-mail: bert.hartnell@smu.ca
More informationIntroduction to Domination Polynomial of a Graph
Introduction to Domination Polynomial of a Graph arxiv:0905.2251v1 [math.co] 14 May 2009 Saeid Alikhani a,b,1 and Yee-hock Peng b,c a Department of Mathematics Yazd University 89195-741, Yazd, Iran b Institute
More informationA note on obtaining k dominating sets from a k-dominating function on a tree
A note on obtaining k dominating sets from a k-dominating function on a tree Robert R. Rubalcaba a,, Peter J. Slater a,b a Department of Mathematical Sciences, University of Alabama in Huntsville, AL 35899,
More informationOn Domination Critical Graphs with Cutvertices having Connected Domination Number 3
International Mathematical Forum, 2, 2007, no. 61, 3041-3052 On Domination Critical Graphs with Cutvertices having Connected Domination Number 3 Nawarat Ananchuen 1 Department of Mathematics, Faculty of
More informationRainbow Connection Number of the Thorn Graph
Applied Mathematical Sciences, Vol. 8, 2014, no. 128, 6373-6377 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.48633 Rainbow Connection Number of the Thorn Graph Yixiao Liu Department
More informationLocating-Domination in Complementary Prisms.
East Tennessee State University Digital Commons @ East Tennessee State University Electronic Theses and Dissertations 5-2009 Locating-Domination in Complementary Prisms. Kristin Renee Stone Holmes East
More informationA Note on an Induced Subgraph Characterization of Domination Perfect Graphs.
A Note on an Induced Subgraph Characterization of Domination Perfect Graphs. Eglantine Camby & Fränk Plein Université Libre de Bruxelles Département de Mathématique Boulevard du Triomphe, 1050 Brussels,
More informationMinimal Spanning Tree From a Minimum Dominating Set
Minimal Spanning Tree From a Minimum Dominating Set M. YAMUNA VIT University School of advanced sciences Vellore, Tamilnadu INDIA myamuna@vit.ac.in K. KARTHIKA VIT University School of advanced sciences
More informationUniversity of Alabama in Huntsville Huntsville, AL 35899, USA
EFFICIENT (j, k)-domination Robert R. Rubalcaba and Peter J. Slater,2 Department of Mathematical Sciences University of Alabama in Huntsville Huntsville, AL 35899, USA e-mail: r.rubalcaba@gmail.com 2 Department
More informationON INTEGER DOMINATION IN GRAPHS AND VIZING-LIKE PROBLEMS. Boštjan Brešar, 1 Michael A. Henning 2 and Sandi Klavžar 3 1.
TAIWANESE JOURNAL OF MATHEMATICS Vol. 10, No. 5, pp. 1317-1328, September 2006 This paper is available online at http://www.math.nthu.edu.tw/tjm/ ON INTEGER DOMINATION IN GRAPHS AND VIZING-LIKE PROBLEMS
More informationarxiv: v1 [math.co] 6 Jan 2017
Domination in intersecting hypergraphs arxiv:70.0564v [math.co] 6 Jan 207 Yanxia Dong, Erfang Shan,2, Shan Li, Liying Kang Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China 2
More informationDominating a family of graphs with small connected subgraphs
Dominating a family of graphs with small connected subgraphs Yair Caro Raphael Yuster Abstract Let F = {G 1,..., G t } be a family of n-vertex graphs defined on the same vertex-set V, and let k be a positive
More informationGraphs with few total dominating sets
Graphs with few total dominating sets Marcin Krzywkowski marcin.krzywkowski@gmail.com Stephan Wagner swagner@sun.ac.za Abstract We give a lower bound for the number of total dominating sets of a graph
More informationOn graphs having a unique minimum independent dominating set
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 68(3) (2017), Pages 357 370 On graphs having a unique minimum independent dominating set Jason Hedetniemi Department of Mathematical Sciences Clemson University
More informationDouble domination in signed graphs
PURE MATHEMATICS RESEARCH ARTICLE Double domination in signed graphs P.K. Ashraf 1 * and K.A. Germina 2 Received: 06 March 2016 Accepted: 21 April 2016 Published: 25 July 2016 *Corresponding author: P.K.
More informationRegular Generalized Star b-continuous Functions in a Bigeneralized Topological Space
International Journal of Mathematical Analysis Vol. 9, 2015, no. 16, 805-815 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.5230 Regular Generalized Star b-continuous Functions in a
More informationTHE RAINBOW DOMINATION NUMBER OF A DIGRAPH
Kragujevac Journal of Mathematics Volume 37() (013), Pages 57 68. THE RAINBOW DOMINATION NUMBER OF A DIGRAPH J. AMJADI 1, A. BAHREMANDPOUR 1, S. M. SHEIKHOLESLAMI 1, AND L. VOLKMANN Abstract. Let D = (V,
More informationSolving Homogeneous Systems with Sub-matrices
Pure Mathematical Sciences, Vol 7, 218, no 1, 11-18 HIKARI Ltd, wwwm-hikaricom https://doiorg/112988/pms218843 Solving Homogeneous Systems with Sub-matrices Massoud Malek Mathematics, California State
More informationRegular Weakly Star Closed Sets in Generalized Topological Spaces 1
Applied Mathematical Sciences, Vol. 9, 2015, no. 79, 3917-3929 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.53237 Regular Weakly Star Closed Sets in Generalized Topological Spaces 1
More informationIntegration over Radius-Decreasing Circles
International Journal of Mathematical Analysis Vol. 9, 2015, no. 12, 569-574 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.47206 Integration over Radius-Decreasing Circles Aniceto B.
More informationResearch Article k-tuple Total Domination in Complementary Prisms
International Scholarly Research Network ISRN Discrete Mathematics Volume 2011, Article ID 68127, 13 pages doi:10.502/2011/68127 Research Article k-tuple Total Domination in Complementary Prisms Adel P.
More informationDiophantine Equations. Elementary Methods
International Mathematical Forum, Vol. 12, 2017, no. 9, 429-438 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.7223 Diophantine Equations. Elementary Methods Rafael Jakimczuk División Matemática,
More informationMappings of the Direct Product of B-algebras
International Journal of Algebra, Vol. 10, 2016, no. 3, 133-140 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.615 Mappings of the Direct Product of B-algebras Jacel Angeline V. Lingcong
More informationDistance in Graphs - Taking the Long View
AKCE J Graphs Combin, 1, No 1 (2004) 1-13 istance in Graphs - Taking the Long View Gary Chartrand 1 and Ping Zhang 2 epartment of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA 1 e-mail:
More informationDiscrete Mathematics. Kernels by monochromatic paths in digraphs with covering number 2
Discrete Mathematics 311 (2011) 1111 1118 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Kernels by monochromatic paths in digraphs with covering
More informationDirect Product of BF-Algebras
International Journal of Algebra, Vol. 10, 2016, no. 3, 125-132 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.614 Direct Product of BF-Algebras Randy C. Teves and Joemar C. Endam Department
More informationOn Regular Prime Graphs of Solvable Groups
International Journal of Algebra, Vol. 10, 2016, no. 10, 491-495 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2016.6858 On Regular Prime Graphs of Solvable Groups Donnie Munyao Kasyoki Department
More informationk-tuple Total Domination in Supergeneralized Petersen Graphs
Communications in Mathematics and Applications Volume (011), Number 1, pp. 9 38 RGN Publications http://www.rgnpublications.com k-tuple Total Domination in Supergeneralized Petersen Graphs Adel P. Kazemi
More informationCharacterization of total restrained domination edge critical unicyclic graphs
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 47 (2010), Pages 77 82 Characterization of total restrained domination edge critical unicyclic graphs Nader Jafari Rad School of Mathematics Institute for Research
More informationGraceful Labeling for Complete Bipartite Graphs
Applied Mathematical Sciences, Vol. 8, 2014, no. 103, 5099-5104 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.46488 Graceful Labeling for Complete Bipartite Graphs V. J. Kaneria Department
More informationMinimizing the Laplacian eigenvalues for trees with given domination number
Linear Algebra and its Applications 419 2006) 648 655 www.elsevier.com/locate/laa Minimizing the Laplacian eigenvalues for trees with given domination number Lihua Feng a,b,, Guihai Yu a, Qiao Li b a School
More informationA Note on Disjoint Dominating Sets in Graphs
Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 43, 2099-2110 A Note on Disjoint Dominating Sets in Graphs V. Anusuya Department of Mathematics S.T. Hindu College Nagercoil 629 002 Tamil Nadu, India
More informationSmall Cycle Cover of 2-Connected Cubic Graphs
. Small Cycle Cover of 2-Connected Cubic Graphs Hong-Jian Lai and Xiangwen Li 1 Department of Mathematics West Virginia University, Morgantown WV 26505 Abstract Every 2-connected simple cubic graph of
More informationDouble domination edge removal critical graphs
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 48 (2010), Pages 285 299 Double domination edge removal critical graphs Soufiane Khelifi Laboratoire LMP2M, Bloc des laboratoires Université demédéa Quartier
More informationACG M and ACG H Functions
International Journal of Mathematical Analysis Vol. 8, 2014, no. 51, 2539-2545 HIKARI Ltd, www.m-hiari.com http://dx.doi.org/10.12988/ijma.2014.410302 ACG M and ACG H Functions Julius V. Benitez Department
More informationReal and Integer Domination in Graphs
Real and Integer Domination in Graphs Wayne Goddard 1 Department of Computer Science, University of Natal, Durban 4041, South Africa Michael A. Henning 1 Department of Mathematics, University of Natal,
More informationDistance between two k-sets and Path-Systems Extendibility
Distance between two k-sets and Path-Systems Extendibility December 2, 2003 Ronald J. Gould (Emory University), Thor C. Whalen (Metron, Inc.) Abstract Given a simple graph G on n vertices, let σ 2 (G)
More informationTransactions on Combinatorics ISSN (print): , ISSN (on-line): Vol. 4 No. 2 (2015), pp c 2015 University of Isfahan
Transactions on Combinatorics ISSN (print): 2251-8657, ISSN (on-line): 2251-8665 Vol. 4 No. 2 (2015), pp. 1-11. c 2015 University of Isfahan www.combinatorics.ir www.ui.ac.ir UNICYCLIC GRAPHS WITH STRONG
More informationMaximum Alliance-Free and Minimum Alliance-Cover Sets
Maximum Alliance-Free and Minimum Alliance-Cover Sets Khurram H. Shafique and Ronald D. Dutton School of Computer Science University of Central Florida Orlando, FL USA 3816 hurram@cs.ucf.edu, dutton@cs.ucf.edu
More informationInternational Journal of Algebra, Vol. 7, 2013, no. 3, HIKARI Ltd, On KUS-Algebras. and Areej T.
International Journal of Algebra, Vol. 7, 2013, no. 3, 131-144 HIKARI Ltd, www.m-hikari.com On KUS-Algebras Samy M. Mostafa a, Mokhtar A. Abdel Naby a, Fayza Abdel Halim b and Areej T. Hameed b a Department
More informationConnected Domination and Spanning Trees with Many Leaves
Connected Domination and Spanning Trees with Many Leaves Yair Caro Douglas B. West Raphael Yuster Abstract Let G = (V, E) be a connected graph. A connected dominating set S V is a dominating set that induces
More informationDisjoint Hamiltonian Cycles in Bipartite Graphs
Disjoint Hamiltonian Cycles in Bipartite Graphs Michael Ferrara 1, Ronald Gould 1, Gerard Tansey 1 Thor Whalen Abstract Let G = (X, Y ) be a bipartite graph and define σ (G) = min{d(x) + d(y) : xy / E(G),
More informationOn the Union of Graphs Ramsey Numbers
Applied Mathematical Sciences, Vol. 8, 2014, no. 16, 767-773 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.311641 On the Union of Graphs Ramsey Numbers I Wayan Sudarsana Combinatorial
More information