Weakly Connected Domination
Koh Khee MengDepartment of MathsNational U of [email protected]
Three Related Dominations
Let G = (V, E) be a connected graph.For , N(v) = the set of neighbors of v, N[v] = N(v) For , ] = • S : dominating set (ds)if i.e., every vertex in V \ S is adjacent to a vertex in S.
V
S
• The domination number of G = γ(G) = min{| S | : S is a ds in G}.Call a minimum ds of G a γ-set of G.
𝛾=2
The n-cube graph, n = 3
[1977] 1st surveypaper by Cockayne & Hedetniemi
• A ds S is called a connected dsif the induced subgraph [S] of G is connected.• The connected domination number of G = (G) = min{| S | : S is a connected ds in G}.Call a minimum connected ds of G a -set of G.
= 4
≤
[1979] Sampathkumar & Walikar
The subgraph weakly induced by S= (
Disjoint union of two K(1, 3)’s
A ds S of G is a weakly connected ds (wcds) of Gif is connected.[𝑺 ]𝒘The weakly connected domination number of G= (G) = min{| S | : S is a wcds in G}.Call a minimum wcds of G a -set of G.
Connected
= 3
() = n ?
4
• () = k
≤
The weakly connected domination was first introduced by Grossman (1997)
● The problem of computing is NP-hard in general.
• ≤ 2
and
• ≤ 2
Relations with other parametersNotation = the independence number of G, = the vertex-covering number of G.Let G be a connected graph of order n ≥ 2. Then • ≤ n/2.
• ≤ =
() = k
= the connectivity of G.
Let G be a connected graph of order n ≥ 2. Then ≤ n -
The equality holds iff or (p-partite, n = 2p).
Sanchis’ works
[1991] Let G be a graph of order n and Then e(G) ≤ .
[2000] Let G be a graph of order n and Then e(G) ≤ + (k – 1).
Grossman asked : How about e(G) if = k ?
Let G be a graph of order n and Then e(G) ≤ .
The extremal graphs are characterized.
TreesIf T is a tree of order n ≥ 2, then = =
The problem of computing is linear for trees.
Let G be a graph. Then = G has a -set G has a S s.t. u
They provide a constructive characterization of trees Tfor which = .
Let T be a tree of order n ≥ 3 and z(T) the number of end-vertices in T. Then
[2004] ≥ (
Let T be a tree of order n ≥ 3 and z(T) the number of end-vertices in T. Then • ≥ (
Let be the family of trees defined recursivelyas follows:
≥ (
(= (21 – 8 + 1)/2= 7
= the family of trees T s.t. = (
Cycle-e-disjoint Graph (Cactus)
connected graph G is a cactus if no two cycles in G have an edge in common;unicyclic if it has exactly one cycle.
For tree T , • ≥ (
Lemanska (2007)
Koh & Xu (2008) Extended the above to unicyclic graphs.
Let G be a cactus of order n ≥ 3; z(G) = number of end-vertices,c(G) = number of cycles,oc(G) = number of odd cycles in G.
Then ≥ ½
RHS = ½(14 – 2 + 1 – 3 – 2)= 4< 5 =
½= ½(14 – 1 + 1 – 3 – 1)= 5=
Cacti for which equality holds are characterized.
A graph G is -stable if (G+e) = (G) edge e in .
For a tree T, TFAE:(1) T is -stable;(2) there is a unique maximum independent set in T;(3) there is a unique -set in T.
G is -unique if it has a unique -set. G is cycle-disjoint if no 2 cycles in G have a vertex in common.The family of -unique cycle-disjoint graphs iscompletely determined.
Applications
Mobile Ad-hoc Networks
Problems• Study (G ) and (G ).
Vizing’s Conjecture (1968)
• Study the criticality of graphs wrt .
• Let G be a connected graph in which every block is either a a cycle or a cycle with a chord. Study (G).
Thank You !
Applications