Page 2 Lecture Notes in Computer Science 2865 Edited by G. Goos ...

Complexity of Connected Components in Evolving Graphs 267h 22h 1232 22h 23K 14h11232v1232h142v 3h2 h21322 32h h24 42v3h413 2 h 3432 v 2 243 2 32h 34322h 33v1v2v31, 4h44v4Fig. 5. Construction for Theorem 2.3.4 Decomposition into o-SCC’sHere we prove the more general result for the case of o-SCC which has a lessstrict definition than SCC. We define the decision problem as follows.o-COMPONENT: Given an evolving digraph G and an integer k, is there ao-SCC of size k?Although SCC’s are a special case of o-SCC’s, the NP-Completeness of COM-PONENT does not directly imply that o-COMPONENT is NP-Complete as well.This is because a possible polynomial time algorithm for o-COMPONENT needonly answer the above decision problem and not identify the o-SCC’s of size k,thus making it difficult to verify if at least one o-SCC of size k is an SCC as well(in other words if the set of h-nodes is empty or not for a particular o-SCC ofsize k). Also, the same digraph G may contain both an SCC (of indeterminatesize) and an o-SCC of size k, soo-COMPONENT would always return “yes”,ignoring the presence or absence of a SCC of size k, thereby leaving COMPO-NENT unsolved. Conversely, since SCC’s are a special case of o-SCC’s, provingo-COMPONENT to be NP-Complete does not directly imply that COMPO-NENT is NP-Complete as well.These arguments entail for an independent proof for the NP-Completenessof o-COMPONENT. Fortunately, however, the same widget utilized for the previousreduction can be applied in the current case, yielding the following results.Theorem 3. o-COMPONENT is in NP.Proof: Same as the proof for Theorem 1.Theorem 4. CLIQUE can be reduced to o-COMPONENT in polynomial time.Proof: Given an undirected graph G =(V,E) and the integer k>3, the samearguments used in the proof of Theorem 2 apply here. Indeed, the same widgetcan be used to reduce CLIQUE to o-SCC, since a SCC is a o-SCC where H = ∅,and in that widget, a max o-SCC is a max SCC, which by Theorem 2 impliesthe reduction from CLIQUE.