CID 2003 Abstracts - Colourings, Independence and Domination

42 abstracts
DOMINATING NUMBERS IN GRAPHS WITH REMOVED
EDGE OR SET OF EDGES
Magdalena Lemańska
Gdańsk University of Technology, Poland
It is known, that the removal of an edge from G cannot decrease a domination
number γ(G) and can increase it by at most one. Thus we can write, that
γ(G) ≤ γ(G − e) ≤ γ(G) + 1 when arbitrary edge is removed. Here we present
similar inequalities for weakly connected domination number γ w and connected
domination number γ c , i.e. we show, that γ w (G) ≤ γ w (G − e) ≤ γ w (G) + 1 and
γ c (G) ≤ γ c (G − e) ≤ γ c (G) + 2 if G and G − e are connected.
We also show that γ w (G) ≤ γ w (G − E p ) ≤ γ w (G) + p − 1 and γ c (G) ≤
γ c (G − E p ) ≤ γ w (G) + 2p − 2 if G and G − E p are connected and E p = E(K p )
where K p ≤ G is the complete subgraph of G.
The distance d(u, v) between two vertices u and v in a connected graph G is
the length of the shortest u−v path in G. A u−v path of length d(u, v) is called
u − v geodesic. A set X ⊂ V is called weakly convex if for every two vertices
a, b ∈ X exists a − b geodesic whose vertices also belong to X and X is called
convex if for every two vertices a, b ∈ X, vertices from every a − b geodesic also
belong to X. We define two new domination parameters γ wcon and γ con .
The weakly convex domination number of G, denoted γ wcon (G), is min{|D| :
D is a minimal weakly convex dominating set of G}, while the convex domination
number of G, denoted γ con (G), is min{|D| : D is a minimal convex
dominating set of G}.
For numbers γ wcon and γ con we show, that differences γ wcon (G)−γ wcon (G− e),
γ wcon (G − e) − γ wcon (G), γ con (G) − γ con (G − e), γ con (G − e) − γ con (G) can be
arbitrarily large.
Keywords: connected dominating number, weakly connected dominating number,
edge removal.
AMS Subject Classification: 05C05, 05C69.
References
[1] T. Haynes, S. Hedetniemi, P. Slater, Fundamentals of domination in
graphs, Dekker, New York (1998).
[2] J. Topp, Domination, independence and irredundance in graphs, Dissertationes
Mathematicae (1995).
[3] J. Dunbar, J. Grossman, S. Hedetniemi, J. Hatting, A. McRae, On weaklyconnected
domination in graphs, Discrete Mathematics 167-168 (1997)
261–269.