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Graphs having distance-n domination number half their order
Graphs having distance-n domination number half their order Miranca Fischermann and Lutz Volkmann Lehrstuhl II f¨ ur Mathematik, RWTH Aachen, Aachen 52056, Germany
For any graph G the vertex set and the order of G are denoted by V (G) and p(G). The distance between two vertices x, y ∈ V (G) in G is denoted by dG (x, y). If e(v) = maxx∈V (G) dG (v, x) for any vertex v ∈ V (G), then rad(G) = minv∈V (G) e(v) and diam(G) = maxv∈V (G) e(v) are the radius and the diameter of G, respectively. The complement G of a graph G has the vertex set V (G) and two vertices are adjacent in G if and only if they are not adjacent in G. Let n be a positive integer and let G be a graph of order p. The distance-n graph Dn (G) has the vertex set V (G) and two vertices are adjacent in Dn (G) if and only if the distance between them in G is n. If diam(G) = 2, then G = D2 (G). A set D ⊆ V (G) is a distance-n dominating set, if every vertex v ∈ V (G) − D has exactly distance n to at least one vertex in D. The minimum cardinality among all distance-n dominating sets is called the distance-n domination number denoted by γ=n (G). For n = 1 a distance-n dominating set D is an ordinary dominating set and the classical domination number γ(G) = γ=1 (G). Unless other indication all notations refer to G (for instance, d(x, y) = dG (x, y)). If diam(G) < n, then the distance-n graph contains no edge and the distance-n domination number γ=n (G) = p(G). Therefore in this paper we assume that diam(G) ≥ n. The distance-n graph, distance-n dominating set, and distance-n domination number were considered by J. W. Boland, T. W. Haynes and L. M. Lawson. Lemma 1 (Boland, Haynes, Lawson [1] 1994) For any graph G and any integer n, γ=n (G) = γ(Dn (G)). Ore’s [4] upper bound p(G)/2 on the domination number applied to the distance-n graph Dn (G) leads to an upper bound on the distance-n domination number. Lemma 2 (Boland, Haynes, Lawson [1] 1994) If Dn (G) has no isolated vertices, then γ=n (G) ≤ p(G)/2. The distance-n graph Dn (G) has no isolated vertices if and only if the radius Preprint submitted to Elsevier Preprint
11 May 1999
rad(G) ≥ n. The bound in Lemma 2 is sharp for every integer n (for instance, G = P2n ). It is interesting to discuss the structure of graphs, for which equality in Lemma 2 holds. Therefore we define the following class of graphs. Let m be an arbitrary integer and let G be a graph of order p with the vertex set {x1 , x2 , . . . , xp }. If Pm denotes a path of m vertices, then the graph G Pm is obtained by taking the graph G and p copies Pm1 , Pm2 , . . . , Pmp of Pm and connecting the vertex xi with one endvertex of Pmi by an edge for all 1 ≤ i ≤ p . Independently of each other Payan and Xuong and Fink, Jacobson, Kinch and Roberts gave a characterization of graphs with γ(G) = γ=1 (G) = p(G)/2. Theorem 3 (Payan, Xuong [5] 1982, Fink, Jacobson, Kinch, Roberts [2] 1985) For a graph G of even order p without isolated vertices, γ(G) = p/2 if and only if the components of G consist of the cycle C4 or the graph H P1 for any connected graph H. Corollary 4 (Boland, Haynes, Lawson [1] 1994) Let n be an integer and let G be a graph of even order p with radius rad(G) ≥ n. Then γ=n(G) = p/2 if and only if the components of Dn (G) consist of the cycle C4 or the graph H P1 for any connected graph H. Using Corollary 4, Boland, Haynes and Lawson characterized all graphs G where G and D2 (G) are connected and the distance-2 domination number γ=2 (G) = p(G)/2. Theorem 5 (Boland, Haynes, Lawson [1] 1994) Let G and D2 (G) be connected of order p ≥ 4. Then γ=2 (G) = p/2 if and only if either diam(G) = 2 and G = H P1 or G = G P3 for any connected graphs H or G . These authors also posed the following conjecture. Conjecture 6 (Boland, Haynes, Lawson [1] 1994) Let G and Dn(G) be connected with an even number of vertices p. Then γ=n(G) = p/2 if and only if G = G P2n−1 for any connected graph G . Remark 7 If a graph G = G P2n−1 for any graph G , then diam(G) ≥ 2n−1. So G = G P2n−1 can only characterize graphs G where γ=n (G) = p/2 and diam(G) ≥ 2n − 1. We have confirmed Conjecture 6 for all graphs with diam(G) ≥ 2n − 1 and n ≥ 2 (Theorem 8). In addition, we give examples of graphs G where G and Dn (G) are connected, γ=n(G) = p/2, but diam(G) < 2n−1 and in consequence G = G P2n−1 (Example 12). Therefore, Conjecture 6 does not hold in general. 2
Theorem 8 Let n ≥ 2 be an integer and let G and Dn (G) be connected of order p such that diam(G) ≥ 2n − 1. Then γ=n(G) = p/2 if and only if G = G P2n−1 for any connected graph G . Remark 9 Theorem 8 yields the case diam(G) = 2 of Theorem 5. For diam(G) = 2 Theorem 5 follows immediately from G = D2 (G) = C4 and Corollary 4. Remark 10 In Theorem 8 it is possible to replace the condition Dn (G) is connected by Dn (G) has no component isomorphic to C4 . (The hypothesis diam(G) ≥ 2n−1 ensures that Dn (G) has no isolated vertices and consequently Lemma 2 implies that γ=n (G) still has the upper bound p(G)/2.) With this new condition Theorem 8 holds also for n = 1, which follows immediately from Theorem 3. For every n ≥ 2 this new condition is weaker than the old one, since Dn (G) = C4 for every graph G and n ≥ 2. With this weaker condition Theorem 8 includes also graphs with disconnected distance-n graph, like P2n . The proofs of Theorem 8 and Remark 10, which also use Corollary 4, can be found in [6]. The following three examples show that Theorem 8 is not valid without the weaker condition of Remark 10. Example 11 Let n ≥ 2 be an arbitrary integer. 1.) G = C4n . 2.) G consists of two disjoint paths x0 x1 . . . xn and y0 y1 . . . yn , along with the three additional edges x0 y0 , x0 y1 , x1 y0 . 3.) G consists of two disjoint paths x0 x1 . . . xn+1 and y0 y1 . . . yn+1, along with the six additional edges x0 y0 , x0 y1 , x0 y2 , x1 y0 , x1 y1 , x2 y0 . All these graphs have diameter diam(G) ≥ 2n − 1 and the distance-n graphs Dn (G) contain at least one component isomorphic to C4 and the remaining components are isomorphic to P2 = P1 P1 . By Corollary 4, we deduce γ=n (G) = p(G)/2, but these graphs obviously do not satisfy G = G P2n−1 . Now we give some examples of graphs G, which satisfy γ=n(G) = p(G)/2 and the conditions of Conjecture 6, but diam(G) < 2n − 1 and consequently G = G P2n−1 . Example 12 1.) Let n ≥ 2 be an arbitrary integer. Regard the graph G consisting of a cycle C2n−1 = x1 x2 . . . x2n−1 x1 , the additional vertices y1 , y2, . . . , y2n−1 and the additional edges y2n−1 x2n−1 , y2n−1 x1 and yi xi , yi xi+1 for every 1 ≤ i ≤ 2n − 2. This graph has diameter n and Dn (G) = C2n−1 P1 . Hence, G and Dn (G) are connected and γ=n(G) = p(G)/2. 3
2.) Let n ≥ 3 be arbitrary. The graph G = C2n−2 P1 has diameter n + 1. For odd integer n we deduce Dn (G) ∼ = G and therefore, G and Dn (G) are connected and γ=n (G) = p/2. For even integer n we get Dn (G) is disconnected, but Dn (G) consists of two components isomorphic to Cn−1 P1 , such that γ=n (G) = p/2. 3.) Let n = 3 and let t ≥ 3 be an arbitrary integer. For i = 1, 2 let Gi be a bipartite graph with the partite sets Xi = {xi1 , xi2 , . . . , xit } and Yi = {yi1 , yi2 , . . . , yit } and the edges xir yis for all 1 ≤ r, s ≤ t with r = s. Regard the graph G consisting of G1 , G2 and of the additional edges, which connect every vertex in X1 with every vertex in X2 . This graph has order p = 4t, diameter n = 3 and D3 (G) = Kt,t P1 . Hence, G and D3 (G) are connected and γ=3 (G) = p(G)/2. Remark 13 Example 12 shows that Conjecture 6 does not hold in general for graphs of diameter lower 2n − 1. For n = 2, 3 we even get that Conjecture 6 does not hold for any diameter lower 2n − 1. (As mentioned above, we regard only graphs G with diameter greater or equal n, otherwise γ=n(G) = p(G).) In addition, for n = 2, 3 we have determined all possible diameters of connected graphs G where Dn (G) has no component isomorphic to C4 and γ=n (G) = p(G)/2. In these cases the possible diameters are n ≤diam(G) ≤ 2n − 1 and diam(G) ≥ 2(2n − 1) + 1.
References [1] J. W. Boland, T. W. Haynes and L. M. Lawson Domination from a Distance. Congr. Numer. 103 (1994), 89–96. [2] J. F. Fink, M. S. Jacobson, L. F. Kinch and J. Roberts On graphs having domination number half their order. Period. Math. Hungar. 16 (1985), 287–293. [3] T. W. Haynes, S. T. Hedetniemi and P. J. Slater Fundamentals of domination in graphs. Marcel Dekker, Inc., New York (1998). [4] O. Ore Theory of Graphs. Amer. Math. Soc. Colloq. Publ. 38 (1962). [5] C. Payan and N. H. Xuong Domination-balanced graphs. J. Graph Theory 6 (1982), 23–32. [6] M. Fischermann and L. Volkmann Graphs having distance-n domination number half their order. Unpublished manuscript (1999).
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For any graph G the vertex set and the order of G are denoted by V (G) and p(G). The distance between two vertices x, y ∈ V (G) in G is denoted by dG (x, y). If e(v) = maxx∈V (G) dG (v, x) for any vertex v ∈ V (G), then rad(G) = minv∈V (G) e(v) and diam(G) = maxv∈V (G) e(v) are the radius and the diameter of G, respectively. The complement G of a graph G has the vertex set V (G) and two vertices are adjacent in G if and only if they are not adjacent in G. Let n be a positive integer and let G be a graph of order p. The distance-n graph Dn (G) has the vertex set V (G) and two vertices are adjacent in Dn (G) if and only if the distance between them in G is n. If diam(G) = 2, then G = D2 (G). A set D ⊆ V (G) is a distance-n dominating set, if every vertex v ∈ V (G) − D has exactly distance n to at least one vertex in D. The minimum cardinality among all distance-n dominating sets is called the distance-n domination number denoted by γ=n (G). For n = 1 a distance-n dominating set D is an ordinary dominating set and the classical domination number γ(G) = γ=1 (G). Unless other indication all notations refer to G (for instance, d(x, y) = dG (x, y)). If diam(G) < n, then the distance-n graph contains no edge and the distance-n domination number γ=n (G) = p(G). Therefore in this paper we assume that diam(G) ≥ n. The distance-n graph, distance-n dominating set, and distance-n domination number were considered by J. W. Boland, T. W. Haynes and L. M. Lawson. Lemma 1 (Boland, Haynes, Lawson [1] 1994) For any graph G and any integer n, γ=n (G) = γ(Dn (G)). Ore’s [4] upper bound p(G)/2 on the domination number applied to the distance-n graph Dn (G) leads to an upper bound on the distance-n domination number. Lemma 2 (Boland, Haynes, Lawson [1] 1994) If Dn (G) has no isolated vertices, then γ=n (G) ≤ p(G)/2. The distance-n graph Dn (G) has no isolated vertices if and only if the radius Preprint submitted to Elsevier Preprint
11 May 1999
rad(G) ≥ n. The bound in Lemma 2 is sharp for every integer n (for instance, G = P2n ). It is interesting to discuss the structure of graphs, for which equality in Lemma 2 holds. Therefore we define the following class of graphs. Let m be an arbitrary integer and let G be a graph of order p with the vertex set {x1 , x2 , . . . , xp }. If Pm denotes a path of m vertices, then the graph G Pm is obtained by taking the graph G and p copies Pm1 , Pm2 , . . . , Pmp of Pm and connecting the vertex xi with one endvertex of Pmi by an edge for all 1 ≤ i ≤ p . Independently of each other Payan and Xuong and Fink, Jacobson, Kinch and Roberts gave a characterization of graphs with γ(G) = γ=1 (G) = p(G)/2. Theorem 3 (Payan, Xuong [5] 1982, Fink, Jacobson, Kinch, Roberts [2] 1985) For a graph G of even order p without isolated vertices, γ(G) = p/2 if and only if the components of G consist of the cycle C4 or the graph H P1 for any connected graph H. Corollary 4 (Boland, Haynes, Lawson [1] 1994) Let n be an integer and let G be a graph of even order p with radius rad(G) ≥ n. Then γ=n(G) = p/2 if and only if the components of Dn (G) consist of the cycle C4 or the graph H P1 for any connected graph H. Using Corollary 4, Boland, Haynes and Lawson characterized all graphs G where G and D2 (G) are connected and the distance-2 domination number γ=2 (G) = p(G)/2. Theorem 5 (Boland, Haynes, Lawson [1] 1994) Let G and D2 (G) be connected of order p ≥ 4. Then γ=2 (G) = p/2 if and only if either diam(G) = 2 and G = H P1 or G = G P3 for any connected graphs H or G . These authors also posed the following conjecture. Conjecture 6 (Boland, Haynes, Lawson [1] 1994) Let G and Dn(G) be connected with an even number of vertices p. Then γ=n(G) = p/2 if and only if G = G P2n−1 for any connected graph G . Remark 7 If a graph G = G P2n−1 for any graph G , then diam(G) ≥ 2n−1. So G = G P2n−1 can only characterize graphs G where γ=n (G) = p/2 and diam(G) ≥ 2n − 1. We have confirmed Conjecture 6 for all graphs with diam(G) ≥ 2n − 1 and n ≥ 2 (Theorem 8). In addition, we give examples of graphs G where G and Dn (G) are connected, γ=n(G) = p/2, but diam(G) < 2n−1 and in consequence G = G P2n−1 (Example 12). Therefore, Conjecture 6 does not hold in general. 2
Theorem 8 Let n ≥ 2 be an integer and let G and Dn (G) be connected of order p such that diam(G) ≥ 2n − 1. Then γ=n(G) = p/2 if and only if G = G P2n−1 for any connected graph G . Remark 9 Theorem 8 yields the case diam(G) = 2 of Theorem 5. For diam(G) = 2 Theorem 5 follows immediately from G = D2 (G) = C4 and Corollary 4. Remark 10 In Theorem 8 it is possible to replace the condition Dn (G) is connected by Dn (G) has no component isomorphic to C4 . (The hypothesis diam(G) ≥ 2n−1 ensures that Dn (G) has no isolated vertices and consequently Lemma 2 implies that γ=n (G) still has the upper bound p(G)/2.) With this new condition Theorem 8 holds also for n = 1, which follows immediately from Theorem 3. For every n ≥ 2 this new condition is weaker than the old one, since Dn (G) = C4 for every graph G and n ≥ 2. With this weaker condition Theorem 8 includes also graphs with disconnected distance-n graph, like P2n . The proofs of Theorem 8 and Remark 10, which also use Corollary 4, can be found in [6]. The following three examples show that Theorem 8 is not valid without the weaker condition of Remark 10. Example 11 Let n ≥ 2 be an arbitrary integer. 1.) G = C4n . 2.) G consists of two disjoint paths x0 x1 . . . xn and y0 y1 . . . yn , along with the three additional edges x0 y0 , x0 y1 , x1 y0 . 3.) G consists of two disjoint paths x0 x1 . . . xn+1 and y0 y1 . . . yn+1, along with the six additional edges x0 y0 , x0 y1 , x0 y2 , x1 y0 , x1 y1 , x2 y0 . All these graphs have diameter diam(G) ≥ 2n − 1 and the distance-n graphs Dn (G) contain at least one component isomorphic to C4 and the remaining components are isomorphic to P2 = P1 P1 . By Corollary 4, we deduce γ=n (G) = p(G)/2, but these graphs obviously do not satisfy G = G P2n−1 . Now we give some examples of graphs G, which satisfy γ=n(G) = p(G)/2 and the conditions of Conjecture 6, but diam(G) < 2n − 1 and consequently G = G P2n−1 . Example 12 1.) Let n ≥ 2 be an arbitrary integer. Regard the graph G consisting of a cycle C2n−1 = x1 x2 . . . x2n−1 x1 , the additional vertices y1 , y2, . . . , y2n−1 and the additional edges y2n−1 x2n−1 , y2n−1 x1 and yi xi , yi xi+1 for every 1 ≤ i ≤ 2n − 2. This graph has diameter n and Dn (G) = C2n−1 P1 . Hence, G and Dn (G) are connected and γ=n(G) = p(G)/2. 3
2.) Let n ≥ 3 be arbitrary. The graph G = C2n−2 P1 has diameter n + 1. For odd integer n we deduce Dn (G) ∼ = G and therefore, G and Dn (G) are connected and γ=n (G) = p/2. For even integer n we get Dn (G) is disconnected, but Dn (G) consists of two components isomorphic to Cn−1 P1 , such that γ=n (G) = p/2. 3.) Let n = 3 and let t ≥ 3 be an arbitrary integer. For i = 1, 2 let Gi be a bipartite graph with the partite sets Xi = {xi1 , xi2 , . . . , xit } and Yi = {yi1 , yi2 , . . . , yit } and the edges xir yis for all 1 ≤ r, s ≤ t with r = s. Regard the graph G consisting of G1 , G2 and of the additional edges, which connect every vertex in X1 with every vertex in X2 . This graph has order p = 4t, diameter n = 3 and D3 (G) = Kt,t P1 . Hence, G and D3 (G) are connected and γ=3 (G) = p(G)/2. Remark 13 Example 12 shows that Conjecture 6 does not hold in general for graphs of diameter lower 2n − 1. For n = 2, 3 we even get that Conjecture 6 does not hold for any diameter lower 2n − 1. (As mentioned above, we regard only graphs G with diameter greater or equal n, otherwise γ=n(G) = p(G).) In addition, for n = 2, 3 we have determined all possible diameters of connected graphs G where Dn (G) has no component isomorphic to C4 and γ=n (G) = p(G)/2. In these cases the possible diameters are n ≤diam(G) ≤ 2n − 1 and diam(G) ≥ 2(2n − 1) + 1.
References [1] J. W. Boland, T. W. Haynes and L. M. Lawson Domination from a Distance. Congr. Numer. 103 (1994), 89–96. [2] J. F. Fink, M. S. Jacobson, L. F. Kinch and J. Roberts On graphs having domination number half their order. Period. Math. Hungar. 16 (1985), 287–293. [3] T. W. Haynes, S. T. Hedetniemi and P. J. Slater Fundamentals of domination in graphs. Marcel Dekker, Inc., New York (1998). [4] O. Ore Theory of Graphs. Amer. Math. Soc. Colloq. Publ. 38 (1962). [5] C. Payan and N. H. Xuong Domination-balanced graphs. J. Graph Theory 6 (1982), 23–32. [6] M. Fischermann and L. Volkmann Graphs having distance-n domination number half their order. Unpublished manuscript (1999).
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