- Email: [email protected]
General distance domination
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 40 (2013) 189–192 www.elsevier.com/locate/endm
General distance domination Elliot Krop 1 Department of Mathematics Clayton State University Morrow, USA
Tony Yaacoub 2 Department of Mathematical Sciences Georgia Southern University Statesboro, USA
Abstract For any graph G = (V, E), a subset S ⊆ V dominates G if all vertices are contained in the closed neighborhood of S, that is N [S] = V . The minimum cardinality over all such S is called the domination number, written γ(G). For any positive integer k, a general k-distance domination function of a graph G is a function f : V → {0, 1, . . . , k} such that every vertex with label 0 is at most distance j − 1 away from a vertex with label j, for 2 ≤ j ≤ k. We show some bounds for this function, produce a Vizing-like bound for the simplest case, and conjecture other more general bounds. Keywords: Domination number, Vizing’s conjecture, Roman domination
1 2
Email: [email protected] Email: [email protected]
1571-0653/$ – see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.endm.2013.05.034
190
1
E. Krop, T. Yaacoub / Electronic Notes in Discrete Mathematics 40 (2013) 189–192
Introduction
All graphs G(V, E) are finite, simple, undirected graphs with vertex set V and edge set E. For a positive integer k, the interval of non-zero integers up to k is written as Ik . For any set of vertices S and non-negative integer j, let N j [S] denote the set of vertices of distance at most j from S. For any graph G = (V, E), a subset S ⊆ V dominates G if N[S] = G, where N[S] denotes the closed neighborhood of S. The minimum cardinality of a subset S of V , so that S dominates G is called the domination number of G and is denoted γ(G). A Roman dominating function of a graph G is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is P adjacent to at least one vertex v for which f (v) = 2. The weight of f is f (V ) = v∈V f (v). The Roman domination number, γR (G), is the minimum weight of a Roman dominating function of G. One can view Roman domination as weighted domination, where domination from larger distances requires greater weight. From this perspective, we can extend the definition to distances greater than 2 by the following Definition 1.1 For any positive integer k, a k-distance domination function of a graph G is a function f : V → {0, 1, . . . , k} such that every vertex with label 0 is at most distance j −P 1 away from a vertex with label j, for 2 ≤ j ≤ k. The weight of f is f (V ) = v∈V f (v). The k-distance domination number, γ{1,...,k} (G), is the minimum weight of a k-distance dominating function of G.
For any non-negative integer i and k-distance dominating function f , let Vif = {v ∈ V (G) : f (v) = i}. We may write Vi = Vif when the function f is clear from context. Besides being a natural extension of domination and Roman domination, study of the above function for Cartesian products of graphs raises interesting questions pertaining to some well-known conjectures discussed below. Graph domination is a well-studied subject in graph theory. The Roman domination problem can be traced back to an attempt at efficient allocation of troops by the Roman emperor Constantine in the fourth century A.C.E. Definition 1.2 The Cartesian product of two graphs G1 (V1 , E1 ) and G2 (V2 , E2 ), denoted by G1 G2 , is the graph with vertex set V1 × V2 and edge set E(G1 G2 ) = {((u1 , v1 ), (u2 , v2 )) : v1 = v2 and (u1 , u2) ∈ E1 , or u1 = u2 and (v1 , v2 ) ∈ E2 }. Perhaps the most popular and elusive conjecture about the domination of
E. Krop, T. Yaacoub / Electronic Notes in Discrete Mathematics 40 (2013) 189–192
191
graphs is due to Vadim G. Vizing (1963) [3], and states that for any graphs G and H γ(GH) ≥ γ(G)γ(H). In section two, we extend some known results (see [1] and [2]) related to this inequality to the distance domination function. In section three, we list a series of Vizing-like conjectures for distance domination numbers and show some sharp examples.
2
Distance domination of cartesian products
Theorem 2.1 For any graphs G and H, and for any positive integer k, let fG , fH , and f be the k-distance dominating functions of minimum weight for G, H, and GH, respectively. Then, ! k X |VjfG | γIk (H) ≤ 2γIk (GH). j=1
By symmetrically applying the result to G and H, we obtain the following result. Corollary 2.2 γIk (G)γIk (H) ≤ 4
(γIk (GH))2 . Pk P f f ( j=1 |Vj G |)( kj=1 |Vj H |)
For any Roman dominating function f , since γ(G) ≤ |V1f | + |V2f |, we have: Corollary 2.3 For any graphs G and H, let fG , fH , and f be the Roman dominating functions of minimum weight for G, H, and GH, respectively. (GH))2 Then, γ(G)γ(H) ≤ 4 γ(γRR(G)γ . R (H)
3
Bounds on distance domination of cartesian products
We believe the bounds in the previous section are far from optimal. In fact, we state Conjecture 3.1 Let G and H be graphs, and let k be a positive integers. If k k+6 γIk (G)γIk (H). If k is odd and is even and at least 2, then γIk (GH) ≥ 2(k+4) k+7 at least 3, then γIk (GH) ≥ 2(k+5) γIk (G)γIk (H). For even k, sharp examples for the above conjecture are graphs of the form Ck+2P2 and likewise, for odd k, we have Ck+3P2 , where Cn denotes a cycle on n vertices.
192
E. Krop, T. Yaacoub / Electronic Notes in Discrete Mathematics 40 (2013) 189–192
Furthermore, we believe that these examples are unique. Thus we state Conjecture 3.2 The above examples are extremal for an upper bound on γIk (GH). Conjecture 3.3 The above examples are unique and extremal for an upper bound on γIk (GH).
References [1] B. Breˇsar, P. Dorbec, W. Goddard, B. Hartnell, M. Henning, S. Klavˇzar, D. Rall, Vizing’s Conjecture: A survey of Recent Results, J. Graph Theory, DOI: 10.1002/jgt.20565. [2] S. Suen, J. Tarr, An Improved Inequality Related to Vizing’s Conjecture, Electron. J. Combin. 19(1) (2012), P8. [3] V. G. Vizing, The Cartesian Product of Graphs, Vychisl. Sistemy 9 (1963), 30–43.
Electronic Notes in Discrete Mathematics 40 (2013) 189–192 www.elsevier.com/locate/endm
General distance domination Elliot Krop 1 Department of Mathematics Clayton State University Morrow, USA
Tony Yaacoub 2 Department of Mathematical Sciences Georgia Southern University Statesboro, USA
Abstract For any graph G = (V, E), a subset S ⊆ V dominates G if all vertices are contained in the closed neighborhood of S, that is N [S] = V . The minimum cardinality over all such S is called the domination number, written γ(G). For any positive integer k, a general k-distance domination function of a graph G is a function f : V → {0, 1, . . . , k} such that every vertex with label 0 is at most distance j − 1 away from a vertex with label j, for 2 ≤ j ≤ k. We show some bounds for this function, produce a Vizing-like bound for the simplest case, and conjecture other more general bounds. Keywords: Domination number, Vizing’s conjecture, Roman domination
1 2
Email: [email protected] Email: [email protected]
1571-0653/$ – see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.endm.2013.05.034
190
1
E. Krop, T. Yaacoub / Electronic Notes in Discrete Mathematics 40 (2013) 189–192
Introduction
All graphs G(V, E) are finite, simple, undirected graphs with vertex set V and edge set E. For a positive integer k, the interval of non-zero integers up to k is written as Ik . For any set of vertices S and non-negative integer j, let N j [S] denote the set of vertices of distance at most j from S. For any graph G = (V, E), a subset S ⊆ V dominates G if N[S] = G, where N[S] denotes the closed neighborhood of S. The minimum cardinality of a subset S of V , so that S dominates G is called the domination number of G and is denoted γ(G). A Roman dominating function of a graph G is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is P adjacent to at least one vertex v for which f (v) = 2. The weight of f is f (V ) = v∈V f (v). The Roman domination number, γR (G), is the minimum weight of a Roman dominating function of G. One can view Roman domination as weighted domination, where domination from larger distances requires greater weight. From this perspective, we can extend the definition to distances greater than 2 by the following Definition 1.1 For any positive integer k, a k-distance domination function of a graph G is a function f : V → {0, 1, . . . , k} such that every vertex with label 0 is at most distance j −P 1 away from a vertex with label j, for 2 ≤ j ≤ k. The weight of f is f (V ) = v∈V f (v). The k-distance domination number, γ{1,...,k} (G), is the minimum weight of a k-distance dominating function of G.
For any non-negative integer i and k-distance dominating function f , let Vif = {v ∈ V (G) : f (v) = i}. We may write Vi = Vif when the function f is clear from context. Besides being a natural extension of domination and Roman domination, study of the above function for Cartesian products of graphs raises interesting questions pertaining to some well-known conjectures discussed below. Graph domination is a well-studied subject in graph theory. The Roman domination problem can be traced back to an attempt at efficient allocation of troops by the Roman emperor Constantine in the fourth century A.C.E. Definition 1.2 The Cartesian product of two graphs G1 (V1 , E1 ) and G2 (V2 , E2 ), denoted by G1 G2 , is the graph with vertex set V1 × V2 and edge set E(G1 G2 ) = {((u1 , v1 ), (u2 , v2 )) : v1 = v2 and (u1 , u2) ∈ E1 , or u1 = u2 and (v1 , v2 ) ∈ E2 }. Perhaps the most popular and elusive conjecture about the domination of
E. Krop, T. Yaacoub / Electronic Notes in Discrete Mathematics 40 (2013) 189–192
191
graphs is due to Vadim G. Vizing (1963) [3], and states that for any graphs G and H γ(GH) ≥ γ(G)γ(H). In section two, we extend some known results (see [1] and [2]) related to this inequality to the distance domination function. In section three, we list a series of Vizing-like conjectures for distance domination numbers and show some sharp examples.
2
Distance domination of cartesian products
Theorem 2.1 For any graphs G and H, and for any positive integer k, let fG , fH , and f be the k-distance dominating functions of minimum weight for G, H, and GH, respectively. Then, ! k X |VjfG | γIk (H) ≤ 2γIk (GH). j=1
By symmetrically applying the result to G and H, we obtain the following result. Corollary 2.2 γIk (G)γIk (H) ≤ 4
(γIk (GH))2 . Pk P f f ( j=1 |Vj G |)( kj=1 |Vj H |)
For any Roman dominating function f , since γ(G) ≤ |V1f | + |V2f |, we have: Corollary 2.3 For any graphs G and H, let fG , fH , and f be the Roman dominating functions of minimum weight for G, H, and GH, respectively. (GH))2 Then, γ(G)γ(H) ≤ 4 γ(γRR(G)γ . R (H)
3
Bounds on distance domination of cartesian products
We believe the bounds in the previous section are far from optimal. In fact, we state Conjecture 3.1 Let G and H be graphs, and let k be a positive integers. If k k+6 γIk (G)γIk (H). If k is odd and is even and at least 2, then γIk (GH) ≥ 2(k+4) k+7 at least 3, then γIk (GH) ≥ 2(k+5) γIk (G)γIk (H). For even k, sharp examples for the above conjecture are graphs of the form Ck+2P2 and likewise, for odd k, we have Ck+3P2 , where Cn denotes a cycle on n vertices.
192
E. Krop, T. Yaacoub / Electronic Notes in Discrete Mathematics 40 (2013) 189–192
Furthermore, we believe that these examples are unique. Thus we state Conjecture 3.2 The above examples are extremal for an upper bound on γIk (GH). Conjecture 3.3 The above examples are unique and extremal for an upper bound on γIk (GH).
References [1] B. Breˇsar, P. Dorbec, W. Goddard, B. Hartnell, M. Henning, S. Klavˇzar, D. Rall, Vizing’s Conjecture: A survey of Recent Results, J. Graph Theory, DOI: 10.1002/jgt.20565. [2] S. Suen, J. Tarr, An Improved Inequality Related to Vizing’s Conjecture, Electron. J. Combin. 19(1) (2012), P8. [3] V. G. Vizing, The Cartesian Product of Graphs, Vychisl. Sistemy 9 (1963), 30–43.