Weakly Connected Domination

Koh Khee MengDepartment of MathsNational U of Singaporematkohkm@nus.edu.sg

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