We perform a systematic study in the computational complexity of the connected variant of three related transversal problems: Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. Just like their original counterparts, these variants are NP-complete for general graphs. A graph @math is @math -free for some graph @math if @math contains no induced subgraph isomorphic to @math . It is known that Connected Vertex Cover is NP-complete even for @math -free graphs if @math contains a claw or a cycle. We show that the two other connected variants also remain NP-complete if @math contains a cycle or claw. In the remaining case @math is a linear forest. We show that Connected Vertex Cover, Connected Feedback Vertex Set, and Connected Odd Cycle Transversal are polynomial-time solvable for @math -free graphs for every constant @math . For proving these results we use known results on the price of connectivity for vertex cover, feedback vertex set, and odd cycle transversal. This is the first application of the price of connectivity that results in polynomial-time algorithms.
Already in the seventies, Garey and Johnson @cite_3 proved that Connected Vertex Cover is -complete even for planar graphs of maximum degree 4. Later, Priyadarsini and Hemalatha @cite_26 strengthened this result to 2-connected planar graphs of maximum degree 4, and Fernau and Manlove @cite_25 strengthened it to planar bipartite graphs of maximum degree 4. Wanatabe, Kajita, and Onaga @cite_15 proved -completeness of Connected Vertex Cover for 3-connected graphs. More recently, Munaro @cite_30 established -completeness of Connected Vertex Cover for line graphs of planar cubic bipartite graphs and for planar bipartite graphs of arbitrarily large girth. Ueno, Kajitani, and Gotoh @cite_5 proved that Connected Vertex Cover can be solved in polynomial time for graphs of maximum degree at most 3 and for trees. Escoffier, Gourv es, and Monnot @cite_2 improved the latter result by showing that Connected Vertex Cover is polynomial-time solvable for chordal graphs. It is known that Vertex Cover is polynomial-time solvable for chordal graphs as well. The same authors proposed to study the complexity of Connected Vertex Cover for other graph classes for which Vertex Cover is polynomial-time solvable.
@cite_30 @cite_26 @cite_3 @cite_2 @cite_5 @cite_15 @cite_25
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The cover time of a graph is a celebrated example of a parameter that is easy to approximate using a randomized algorithm, but for which no constant factor deterministic polynomial time approximation is known. A breakthrough due to Kahn, Kim, Lovasz and Vu [25] yielded a (log logn)2 polynomial time approximation. We refine the upper bound of [25], and show that the resulting bound is sharp and explicitly computable in random graphs. Cooper and Frieze showed that the cover time of the largest component of the Erdős–Renyi random graph G(n, c n) in the supercritical regime with c > 1 fixed, is asymptotic to ϕ(c)nlog2n, where ϕ(c) → 1 as c ↓ 1. However, our new bound implies that the cover time for the critical Erdős–Renyi random graph G(n, 1 n) has order n, and shows how the cover time evolves from the critical window to the supercritical phase. Our general estimate also yields the order of the cover time for a variety of other concrete graphs, including critical percolation clusters on the Hamming hypercube 0, 1 n, on high-girth expanders, and on tori ℤdn for fixed large d. This approach also gives a simpler proof of a result of Aldous [2] that the cover time of a uniform labelled tree on k vertices is of order k3 2. For the graphs we consider, our results show that the blanket time, introduced by Winkler and Zuckerman [45], is within a constant factor of the cover time. Finally, we prove that for any connected graph, adding an edge can increase the cover time by at most a factor of 4. We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5 3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs). Finally, dealing with hypergraphs, we study the complexity and the approximability of two natural generalizations. Important variants of theVERTEX COVER problem (among others, CONNECTED VERTEX COVER, CAPACITATED VERTEX COVER, and MAXIMUM PARTIAL VERTEX COVER) have been intensively studied in terms of polynomial-time approximability. By way of contrast, their parameterized complexity has so far been completely open. We close this gap here by showing that, with the size of the desired vertex cover as the parameter, CONNECTED VERTEX COVER and CAPACITATED VERTEX COVER are both fixed-parameter tractable while MAXIMUM PARTIAL VERTEX COVER is W[1]-complete. This answers two open questions from the literature. The results extend to several closely related problems. Interestingly, although the considered variants of VERTEX COVER behave very similar in terms of constant factor approximability, they display a wide range of different characteristics when investigating their parameterized complexities. We continue research into a well-studied family of problems that ask if the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. The problem is known to be polynomial-time solvable if k=0 (bipartite graphs) and NP-complete if k=1 (near-bipartite graphs) even for graphs of diameter 4, as shown by Yang and Yuan, who also proved polynomial-time solvability for graphs of diameter 2. We show that recognizing near-bipartite graphs of diameter 3 is NP-complete resolving their open problem. To answer another open problem, we consider graphs of maximum degree D on n vertices. We show how to find A and B in O(n) time for k=1 and D=3, and in O(n^2) time for k >= 2 and D >= 4. These results also provide an algorithmic version of a result of Catlin [JCTB, 1979] and enable us to complete the complexity classification of another problem: finding a path in the vertex colouring reconfiguration graph between two given k-colourings of a graph of bounded maximum degree.