Discusión de resultados de la consulta a expertos

The idea of decreasing graphs is an adaptation of the augmenting graph technique used to solve the maximum independent set problem in various classes of graphs. In particular, this technique was used by Edmonds to solve the maximum matching problem, which is equivalent to the maximum independent set problem in the class of line graphs. See [113, 100, 114] for more applications of this technique. The idea of augmenting graphs consists in step-by-step increasing a current solution until a maximum independent set is obtained. In the case of the independent dominating setproblem we iteratively decrease a current solution. The main idea of this approach can be described as follows.

Let G be a vertex-weighted graph and I an independent dominating set in

G. We denote the weight of a set U ⊆ V(G) by ω(U). If G contains a bipartite induced subgraph B= (B1, B2, F) such that

(i) B1∩I =∅ and B2 ⊆I, (ii) ω(B1)< ω(B2),

(iii) B1∪(I\B2) is an independent dominating set,

thenI is not a minimum weight independent dominating set inG, sinceB1∪(I\B2) is an independent dominating set of smaller weight. On the other hand, ifI is not a minimum weight independent dominating set and if J is an arbitrary minimum weight independent dominating set inG, then B1 := J\I and B2 :=I \J induce a bipartite graph satisfying (i), (ii), (iii). We call a bipartite graph B satisfying (i), (ii), (iii) a decreasing graph for I, and we call the difference ω(B2)−ω(B1) thedecrementof B. Observe that by definition the decrement is a strictly positive number. If there is a decreasing graph forI, we also say thatI admits a decreasing graph.

According to the above discussion, an independent dominating set in a graph

Gis of minimum weight if and only ifI admits no decreasing graph. Let us observe that a decreasing graph may, in general, be disconnected. However, if we deal with

P5-free graphs, we may restrict ourselves toconnecteddecreasing graphs, as we show in the following lemma.

Lemma 66. Let G be a P5-free vertex-weighted graph and I an independent dom- inating set in G. Then I is an independent dominating set of minimum weight if and only if it admits no connecteddecreasing graph.

Proof. If I is a minimum weight independent dominating set, then it admits no decreasing graphs (including connected decreasing graphs), which proves the lemma in one direction. To prove it in the other direction, assumeI admits no connected decreasing graph, and suppose by contradiction thatIis not a minimum weight inde- pendent dominating set. Then there must exist a decreasing graphB = (B1, B2, F) with at least two connected components. Each component of B must contain at least two vertices, since otherwise eitherI orB1∪(I\B2) is not a dominating set. Among the components of B there must exist at least one component with strictly positive decrement. Let B0 = (B₁0, B₂0, F0) be such a component. By assumption,

B₁0 ∪(I\B₂0) is not a dominating set inG, since otherwiseB0 would be a connected decreasing graph. Therefore, there must exist a vertex u that does not belong to

B₁0 ∪(I\B₂0) and has no neighbour in this set. We observe that

• udoes not belong to the setB0₂ (and hence toI), since every vertex ofB₂0 has a neighbour inB₁0 due to the connectedness ofB0,

• uhas a neighbour b0₂ inB0₂, sinceI is dominating,

• udoes not belong toB1, since the only vertices ofB1 that have neighbours in

B₂0 are those in the set B₁0, butu does not belongB₁0,

• uhas a neighbour b00₁ inB1\B10, sinceB1∪(I\B2) is a dominating set. Since each connected component of B has at least two vertices, b0₂ must have a neighbourb0₁ ∈B₁0, whileb00₁ must have a neighbour b00₂ ∈B2\B20. But then vertices

b0₁, b0₂, b00₁, b00₂, u induce a P5 in G. This contradiction completes the proof of the lemma.

Let us now apply Lemma 66 to solveweighted independent domination

in the class of (P5, K1,p)-free graphs for any fixed value ofp. We emphasize that in

the current version the solution applies only to graphs of polynomial weight, i.e. we assume that the total weight of the input graph is bounded by a polynomial in the number of vertices.

Theorem 67.Theweighted independent dominating setproblem for(P5, K1,p)-

free graphs of polynomial weight can be solved in polynomial time for each fixed value of p.

Proof. LetGbe a (P5, K1,p)-free graph andI an arbitrary independent dominating

set inG. By Lemma 66,I is a minimum weight independent dominating set if and only if it admits no connected decreasing graph. It is well-known (and not difficult

to see) that a connectedP5-free bipartite graph is 2K2-free. Therefore, the vertices in each part of a connectedP5-free bipartite graph can be ordered under inclusion of their neighbourhoods, and hence any vertex with a maximal neighbourhood is adjacent to all vertices in the opposite part. This implies that if a connectedP5-free bipartite graph isK1,p-free, then each part of the graph has at mostp−1 vertices.

Thus,I is a minimum weight independent dominating set if and only if it admits no connected decreasing graph with at most 2p−2 vertices. If such a graph exists for

I, we obtained an independent dominating set of smaller weight by exchanging the parts of the graph. Since the weights of the vertices are bounded by a polynomial, in polynomially many iterations we obtain an independent dominating set of minimum weight. Each iteration can be done inO(n2p) time, and hence the overall complexity of the procedure is bounded by a polynomial.

6.6

Conclusion

In this chapter we first gave an algorithm which generates maximal independent sets in 2K2-free graphs, inspired by Farber’s proof that such graphs have only polynomi- ally many maximal independent sets and went on to extend the algorithm to larger classes of graphs. In particular, we solved the problem in the class ofP2+P3-free graphs, correcting a mistake in [87], as well as in the class of (P5,2P3)-free graphs. We considered a number of subclasses ofP5-free graphs obtained by forbidding one additional induced subgraph, making use of modular decomposition and so-called decreasing graphs to give a collection of polynomial time results. These results are extended from solutions in (P5, F)-free graphs to (P5, F +pK2)-free graphs and (P5, F∗)-free graphs, whereF∗ is obtained fromF by adding three vertices inducing aP3 such that one end vertex dominates the vertices ofF.

Notice that all results in this chapter deal with classes defined by finitely many forbidden induced subgraphs. A useful tool to study complexity of algorithmic problems on finitely defined classes is the notion of boundary properties of graphs. This notion was introduced by Alekseev in [62] for themaximum independent set

problem and then was extended in [113, 115, 15] to other problems. The importance of this notion is due to the fact that an NP-hard algorithmic graph problem Π can be solved in polynomial time in a finitely defined classX if and only ifX contains none of the boundary classes for Π.

At present, only two boundary classes are known for independent dom- ination: the class S and the class T of line graphs of graphs in S. Also, it is

in this class. Therefore, the class of SAT-graphs must contain a boundary class for the problem. However, neither S nor T is a subclass of SAT-graphs, and hence, there must exist at least one more boundary class for the problem. We discover this class in the concluding part of the thesis by exploiting the relationship between

Part III

Satisfiability

and

Chapter 7

Boundary Properties of the

Satisfiability Problem