Domination Subdivision Number Research Articles

Perfect Matchings and K 4 3 -Tilings in Hypergraphs of Large Codegree

For a k-graph F, let t l (n, m, F) be the smallest integer t such that every k-graph G on n vertices in which every l-set of vertices is included in at least t edges contains a collection of vertex-disjoint F-subgraphs covering all but at most m vertices of G. Let K m k denote the complete k-graph on m vertices. The function $$t_{k-1} (kn, 0, K_k^k)$$(i.e. when we want to guarantee a perfect matching) has been previously determined by Kuhn and Osthus [9] (asymptotically) and by Rodl, Rucinski, and Szemeredi [13] (exactly). Here we obtain asymptotic formulae for some other l. Namely, we prove that for any $$k \ge 4$$and $$k/2 \le l \le k-2$$, $$ t_l(kn, 0, K_k^k) = \left(\frac{1}{2} + o(1)\right) {kn\choose k-l} $$is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G. The domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Haynes et al. (Discussiones Mathematicae Graph Theory 21 (2001) 239-253) conjectured that $${\rm sd}_{\gamma} (G) \le \delta(G) + 1$$for any graph G with $$\delta(G) \ge 2$$. In this note we first give a counterexample to this conjecture in general and then we prove it for a particular class of graphs.

Disproof of a Conjecture on the Subdivision Domination Number of a Graph

A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of $$V(G) \setminus S$$ is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G. The domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Haynes et al. (Discussiones Mathematicae Graph Theory 21 (2001) 239-253) conjectured that $${\rm sd}_{\gamma} (G) \le \delta(G) + 1$$ for any graph G with $$\delta(G) \ge 2$$. In this note we first give a counterexample to this conjecture in general and then we prove it for a particular class of graphs.

Domination subdivision numbers of trees

A set S of vertices of a graph G = ( V , E ) is a dominating set if every vertex of V ( G ) ∖ S is adjacent to some vertex in S . The domination number γ ( G ) is the minimum cardinality of a dominating set of G . The domination subdivision number sd γ ( G ) is the minimum number of edges that must be subdivided in order to increase the domination number. Velammal showed that for any tree T of order at least 3, 1 ≤ sd γ ( T ) ≤ 3 . In this paper, we give two characterizations of trees whose domination subdivision number is 3 and a linear algorithm for recognizing them.

Upper bounds for the Roman domination subdivision number of a graph

A Roman dominating function of a graph G is a labeling f: V(G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of Σv∊V(G) f(v) over such functions. The Roman domination subdivision number sdγR (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the Roman domination number. In this paper, we establish upper bounds for the Roman domination subdivision number of graphs.

Weakly connected domination subdivision numbers

A set D of vertices in a graph G = (V, E) is a weakly connected dominating set of G if D is dominating in G and the subgraph weakly induced by D is connected. The weakly connected domination number of G is the minimum cardinality of a weakly connected dominating set of G. The weakly connected domination subdivision number of a connected graph G is the minimum number of edges that must be subdivided (where each egde can be subdivided at most once) in order to increase the weakly connected domination number. We study the weakly connected domination subdivision numbers of some families of graphs.

Total domination and total domination subdivision number of a graph and its complement

A set S of vertices of a graph G = ( V , E ) with no isolated vertex is a total dominating set if every vertex of V ( G ) is adjacent to some vertex in S. The total domination number γ t ( G ) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd γ t ( G ) is the minimum number of edges that must be subdivided in order to increase the total domination number. We consider graphs of order n ⩾ 4 , minimum degree δ and maximum degree Δ . We prove that if each component of G and G ¯ has order at least 3 and G , G ¯ ≠ C 5 , then γ t ( G ) + γ t ( G ¯ ) ⩽ 2 n 3 + 2 and if each component of G and G ¯ has order at least 2 and at least one component of G and G ¯ has order at least 3, then sd γ t ( G ) + sd γ t ( G ¯ ) ⩽ 2 n 3 + 2 . We also give a result on γ t ( G ) + γ t ( G ¯ ) stronger than a conjecture by Harary and Haynes.

Characterization of double domination subdivision number of trees

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V ( G ) is a double dominating set of G if S dominates every vertex of G at least twice. The double domination number dd ( G ) is the minimum cardinality of a double dominating set of G. The double domination subdivision number sd dd ( G ) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the double domination number. In this paper first we establish upper bounds on the double domination subdivision number for arbitrary graphs in terms of vertex degree. Then we present several different conditions on G which are sufficient to imply that sd dd ( G ) ⩽ 3 . We also prove that 1 ⩽ sd dd ( T ) ⩽ 2 for every tree T, and characterize the trees T for which sd dd ( T ) = 2 .

Total domination subdivision numbers of trees

A set S of vertices in a graph G is a total dominating set of G if every vertex is adjacent to a vertex in S. The total domination number γ t ( G ) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd γ t ( G ) of a graph G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the total domination number. Haynes et al. (J. Combin. Math. Combin. Comput. 44 (2003) 115) showed that for any tree T of order at least 3, 1 ⩽ sd γ t ( T ) ⩽ 3 . In this paper, we give a constructive characterization of trees whose total domination subdivision number is 3.

Subdivision Number of Graphs and Falsity of a conjecture

Abstract The domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (where each edge in G can be subdivided atmost once in order to increase the domination number). Arumugam made an interesting conjecture for arbitrary graphs namely for any graph G of order n ≥ 3,3 ≥ sdγ ( G ) ≥ 1. Haynes Hedetniemi, Hedetniemi Jacobs, Knosely and Van der Merwe gave a counter example to the above conjecture by showing that sdγ ( G ) = 4 for the graph K t × K t where t ≥ 4 and modified the conjecture as 4 ≥ sdr ( G ) ≥ 1. In this paper we give an example of a graph with sdγ ( G ) = 5. Restrained subdivision number sdγr(G) is also defined and we show that there exists a graph with sdγr ( G ) = 5.

Domination Subdivision Numbers

A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of V − S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam conjectured that 1 ≤ sdγ(G) ≤ 3 for any graph G. We give a counterexample to this conjecture. On the other hand, we show that sdγ(G) ≤ γ(G)+1 for any graph G without isolated vertices, and give constant upper bounds on sdγ(G) for several families of graphs.

Domination and independence subdivision numbers of graphs

The domination subdivision number sd∞(G) of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upper bound for the domination subdivision number for any graph G in terms of the minimum degrees of adjacent vertices in G. We then define the independence subdivision number sdfl(G) to equal the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the independence number. We show that for any graph G of order n ‚ 2, either G = K1;m and sdfl(G) = m, or 1 • sdfl(G) • 2. We also characterize the graphs G for which sdfl(G) = 2.