independent domination in finitely defined classes of graphs polynomial algorithms

Independent domination in finitely defined classes of graphs:

Polynomial algorithms

Vadim Lozin, Raffaele Mosca, Christopher Purcell

Discrete Applied Mathematics 182 (2015) 2–14

報告者 : 陳政謙

Outline

• Related works• Independent domination in P2 + P3-free graphs• References

Related worksJournal Topic Author Result

Theoretical Computer Science 301 (2003) 271 – 284

Independent domination in finitely defined classes of graphs

R. Boliac, V. Lozin Some sufficient conditions for the independent domination problem are to be NP-hard in a finitely defined class of graphs.

Information Processing Letters 36 (1990) 231-236

Domination in convex and chordal bipartite graphs

Peter Damaschke, Haiko Müller, Dieter Kratsch

It is shown by a reduction from 3SAT that independent dominating set remains NP-complete when restricted to chordal bipartite graphs.


Operations Research Letters 1 (1982) 134-138

Independent domination in chordal graphs

Martin Farber There is a linear algorithm to solve the independent domination problem in chordal graphs.

Discrete Applied Mathematics 143 (2004) 351 – 352

The weighted independent domination problem is NP-completefor chordal graphs

Gerard J. Chang This paper shows that the weighted independent domination problem is NP-complete for chordal graphs.


Discrete Mathematics 73 (1989) 249-260

On diameters and radii of bridged graphs

Martin Farber This paper proved that 2K2-free graphs have polynomially many maximal independent sets.

The independent dominating set

The independent domination problem

Pn graphs

P2

P3

P2 + P3

P2 + P3-free graphs

Algorithm: Generation-1

S = {{v1}}u({v1}) = v2

v1 v3 v4 v5v2


u({v2}) = v1

S = {{v1}, {v2}}u({v1}) = v2

v1 v4 v5v2 v3


v1 v4 v5v2 v3

u({v2}) = v1

S = {{v1}, {v2}}u({v1}) = v2


v1 v4 v5v2 v3

u({v2}) = v1

S = {{v1, v3}, {v2}}u({v1, v3}) = v2


u({v3}) = v2

v1 v5v2 v3

u({v2}) = v1

S = {{v1, v3}, {v2}, {v3}}u({v1, v3}) = v2

v4


v1 v5v2 v3 v4

u({v3}) = v2u({v2}) = v1

S = {{v1, v3}, {v2}, {v3}}u({v1, v3}) = v2


v1 v5v2 v3 v4

u({v3}) = v2u({v2, v4}) = v1

S = {{v1, v3}, {v2, v4}, {v3}}u({v1, v3}) = v2


v1 v5v2 v3 v4

u({v3}) = v2u({v2, v4}) = v1

S = {{v1, v3}, {v2, v4}, {v3}, {v1, v4}}u({v1, v3}) = v2 u({v1, v4}) = v3


v1 v2 v3 v4

u({v3}) = v2u({v2, v4}) = v3

S = {{v1, v3}, {v2, v4}, {v3}, {v1, v4}}u({v1, v3}) = v2 u({v1, v4}) = v3

v5


v1 v2 v3 v4

u({v3}) = v2u({v2, v4}) = v3

S = {{v1, v3, v5}, {v2, v4}, {v3}, {v1, v4}}u({v1, v3, v5}) = v2 u({v1, v4}) = v3

v5


v1 v2 v3 v4

u({v3, v5}) = v2u({v2, v4}) = v3

S = {{v1, v3, v5}, {v2, v4}, {v3, v5}, {v1, v4}}u({v1, v3, v5}) = v2

u({v1, v4}) = v3

v5


v1 v2 v3 v4

u({v3, v5}) = v2u({v2, v4}) = v3

S = {{v1, v3, v5}, {v2, v4}, {v3, v5}, {v1, v4}, {v1, v2, v5}}u({v1, v3, v5}) = v2

u({v1, v4}) = v3

v5

u({v1, v2 , v5}) = v4


v1 v2 v3 v4

u({v3, v5}) = v2u({v2, v4}) = v3

S = {{v1, v3, v5}, {v2, v4}, {v3, v5}, {v1, v4}, {v1, v2, v5}}u({v1, v3, v5}) = v4

u({v1, v4}) = v3

v5

u({v1, v2 , v5}) = v4

Lemma 3• For a graph G with n vertices, Algorithm Generation-1 runs in

time O(n5) and the family S produced by this algorithm contains O(n3) subsets of V(G).

If a set H S was created in Step 2.2∈

u

G

v

AG({v, u, w})

w

H := {v, w} A∪ G ({v, u, w})

If a set H S was created in Step 2.1∈

G

v

AG({v, u})u

We denote W the set of neighbors of u each of which is not dominatedby every maximal independent set in G[H].

W

H := {v} A∪ G ({v, u})

Proposition 4• A set H created in Step 2.1 contains an independent set

dominating G if and only if H – H0 contains an independent set dominating W.

Lemma 5• If ab is an edge in G[H – H0], then W N(a) N(b).⊆ ∪

G

v

AG({v, u})u

W

a b

The proof of Lemma 5• If ab is an edge in G[H – H0], then W N(a) N(b).⊆ ∪

G

v

AG({v, u})u

W

a b

The partition of cliques in G[H – H0]

• Lemma 5 shows that if Q = {q1, ... , qp} is a component (clique) in G[H – H0] and Wi = W ∩ AG(qi), then {W1, ... , Wp} is a partition of W. We denote this partition by P(Q).

q1

q2 q3

v1

v2

v3

W

P(Q) = {{v1, v2}, {v3}}

Q

Lemma 6• The set H – H0 contains a maximal independent set

dominating W if and only if there is an element (Y1, . . . , Yt) ∈P(Q1) × … × P(Qt) such that Y1 ∩ … ∩ Yt = .∅P(Q1) = {{v1}, {v2, v3}}P(Q2) = {{v1, v2}, {v3}}

{v1} ∩ {v3} = ∅

The proof of Lemma 6

G

v

H – H0u

W

Y1

Y2

Q1

Q2


G

v

H – H0u

W1

Q1

Q2

2


• Therefore, I dominates W if and only if … ∪ ∪= W.

• By De Morgan’s law, this holds if and only if Y1 ∩ … ∩ Yt = .∅

Lemma 7• Given a set W of n elements and a number of partitions P1, . . .

,Pt of W, one can check if there is an element (Y1, . . . , Yt) P∈ 1 × …× Pt such Y1 ∩…∩ Yt = in O(n∅ 2) time.

P1

P2

{v1} {v2, v3}

W = {v1, v2, v3}P1 = {{v1}, {v2, v3}}P2 = {{v1, v2}, {v3}}

{v1, v2, v3}

{v1} { }∅ {v2} {v3}

Theorem 8• Given a P2 + P3-free graph G with n vertices, one can find an

independent dominating set of minimum cardinality in G in O(n5) time.

By Lemma 3, the time complexity of Algorithm Generation-1 is O(n5).

By Proposition 4, Lemmas 6 and 7, the problem of determining if H S contains a maximal independent set dominating G can be ∈solved in O(n2) time.

Therefore, an independent dominating set of minimum cardinality in G can be found in O(n5) time.

References

independent domination in finitely defined classes of graphs polynomial algorithms

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