Moo Young Sohn

Bounds on the locating-total domination number of a tree

In this paper, we continue the study of locating-total domination in graphs, introduced by Haynes et al. [T.W. Haynes, M.A. Henning, J. Howard, Locating and total dominating sets in trees, Discrete Applied Mathematics 154 (8) (2006) 1293-1300]. A total dominating set S in a graph G=(V,E) is a locating-total dominating set of G if, for every pair of distinct vertices u and v in V-S, NG(u)@?S NG(v)@?S. The minimum cardinality of a locating-total dominating set is the locating-total domination number @ct^L(G). We show that, for a tree T of order n>=3 with l leaves and s support vertices, n+l+12-s@?@ct^L(T)@?n+l2. Moreover, we constructively characterize the extremal trees achieving these bounds.

Bondage number of the discrete torus C n ×C 4

The bondage number b(G) of a graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with a domination number greater than the domination number of G. In this paper, we show that the bondage number of the Cartesian product CnxC4 of two cycles Cn(n>=4) and C4 is equal to 4, i.e., b(CnxC4)=4 for any n>=4.

Paired-domination in inflated graphs

The inflation G1 of a graph G with n(G) vertices and m(G) edges is obtained from G by replacing every vertex of degree d of G by a clique Kd. A set S of vertices in a graph G is a paired dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired domination number λp(G) is the minimum cardinality of a paired dominating set of G. In this paper, we show that if a graph G has a minimum degree δ(G) ≥ 2, then n(G) ≤ λp(G1) ≤ 4m(G)/[δ(G) + 1], and the equality λp(G1)=n(G) holds if and only if G has a perfect matching. In addition, we present a linear time algorithm to compute a minimum paired-dominating set for an inflation tree.

The algorithmic complexity of mixed domination in graphs

Let G=(V,E) be a simple graph with vertex set V and edge set E. A subset [emailxa0protected][emailxa0protected]?E is a mixed dominating set if every element [emailxa0protected]?([emailxa0protected]?E)@?W is either adjacent or incident to an element of W. The mixed domination problem is to find a minimum mixed dominating set of G. In this paper we first prove that a connected graph is a tree if and only if its total graph is strongly chordal, and thus we obtain a polynomial-time algorithm for this problem in trees. Further we design another linear-time labeling algorithm for this problem in trees. At the end of the paper, we show that the mixed domination problem is NP-complete even when restricted to split graphs, a subclass of chordal graphs.

Bipartite covering graphs

Abstract In this paper we study when a bipartite graph is a covering of a non-bipartite graph. We give a characterization of all bipartite coverings in terms of factoring the covering map through the canonical double covering. We also consider regular bipartite coverings described in terms of voltage assignments. We give an algebraic characterization of such coverings involving the subgroup generated by voltages on closed walks of even length. This allows us to count the number of regular bipartite coverings for orders which are twice a prime.

Minus domination number in k-partite graphs

A function f defined on the vertices of a graph G=(V,E),f:V?{-1,0,1} is a minus dominating function if the sum of its values over any closed neighborhood is at least one. The weight of a minus dominating function is f(V)=?v?Vf(v). The minus domination number of a graph G, denoted by γ-(G), equals the minimum weight of a minus dominating function of G. In this paper, a sharp lower bound on γ- of k-partite graphs is given. The special case k=2 implies that a conjecture proposed by Dunbar et al. (Discrete Math. 199(1999) 35) is true.

Bartholdi zeta functions of graph bundles having regular fibers

As a continuation of computing the Bartholdi zeta function of a regular covering of a graph by Mizuno and Sato in J. Combin. Theory Ser. B 89 (2003) 27, we derive in this paper some computational formulae for the Bartholdi zeta functions of a graph bundle and of any (regular or irregular) covering of a graph. If the fiber is a Schreier graph or it is regular and the voltages to derive the bundle or the covering lie in an Abelian group, then the formulae can be simplified. As a byproduct, the Bartholdi zeta functions of Schreier graphs, Cayley graphs and the cartesian product of a graph and a regular graph are obtained.

On the existence problem of the total domination vertex critical graphs

The existence problem of the total domination vertex critical graphs has been studied in a series of articles. We first settle the existence problem with respect to the parities of the total domination number m and the maximum degree @D: for even m except m=4, there is no m-@ct-critical graph regardless of the parity of @D; for m=4 or odd m>=3 and for even @D, an m-@ct-critical graph exists if and only if @D>=2@?m-12@?; for m=4 or odd m>=3 and for odd @D, if @D>=2@?m-12@?+7, then m-@ct-critical graphs exist, if @D =9. As the previously known result for m=3, we also show that for @D(G)=3,5,7, there is no 4-@ct-critical graph of order @D(G)+4. On the contrary, it is shown that for odd m>=9 there exists an m-@ct-critical graph for all @D>=m-1.

Isoperimetric numbers of graph bundles

The main aim of this paper is to give some upper and lower bounds for the isoperimetric numbers of graph coverings or graph bundles, with exact values in some special cases. In addition, we show that the isoperimetric number of any covering graph is not greater than that of the base graph. Mohars theorem for the isoperimetric number of the cartesian product of a graph and a complete graph can be extended to a more general case: The isoperimetric numberi(G × K2n) of the cartesian product of any graphG and a complete graphK2n on even vertices is the minimum of the isoperimetric numberi(G) andn, and it is also a sharp lower bound of the isoperimetric numbers of all graph bundles over the graphG with fiberK2n. Furthermore, ifn ≥ 2i(G) then the isoperimetric number of any graph bundle overG with fibreKn is equal to the isoperimetric numberi(G) ofG.

CHARACTERISTIC POLYNOMIALS OF SOME WEIGHTED GRAPH BUNDLESAND ITS APPLICATION TO LINKS

In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weighted K2 (K2)- bundles over a weighted graphF can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs are F As an application, we compute the signature of a link whose corresponding weighted graph is a double covering of that of a given link.