Cycle Transversal Research Articles

Clique cycle transversals in distance-hereditary graphs

A cycle transversal of a graph G is a subset T⊆V(G) such that T∩V(C)≠∅ for every cycle C of G. A clique cycle transversal, or cct for short, is a cycle transversal which is a clique. Recognizing graphs which admit a cct can be done in polynomial time; however, no structural characterization of such graphs is known. We characterize distance-hereditary graphs admitting a cct in terms of forbidden induced subgraphs. This extends similar results for chordal graphs and cographs.

Preprocessing subgraph and minor problems: When does a small vertex cover help?

We prove a number of results around kernelization of problems parameterized by the size of a given vertex cover of the input graph. We provide three sets of simple general conditions characterizing problems admitting kernels of polynomial size. Our characterizations not only give generic explanations for the existence of many known polynomial kernels for problems like q-Coloring, Odd Cycle Transversal, Chordal Deletion, η-Transversal, or Long Path, parameterized by the size of a vertex cover, but also imply new polynomial kernels for problems like F-Minor-Free Deletion, which is to delete at most k vertices to obtain a graph with no minor from a fixed finite set F. While our characterization captures many interesting problems, the kernelization complexity landscape of parameterizations by vertex cover is much more involved. We demonstrate this by several results about induced subgraph and minor containment testing.

Clique cycle transversals in graphs with few P₄'s

Graph Theory A graph is extended P4-laden if each of its induced subgraphs with at most six vertices that contains more than two induced P4's is 2K2,C4-free. A cycle transversal (or feedback vertex set) of a graph G is a subset T ⊆ V (G) such that T ∩ V (C) 6= ∅ for every cycle C of G; if, in addition, T is a clique, then T is a clique cycle transversal (cct). Finding a cct in a graph G is equivalent to partitioning V (G) into subsets C and F such that C induces a complete subgraph and F an acyclic subgraph. This work considers the problem of characterizing extended P4-laden graphs admitting a cct. We characterize such graphs by means of a finite family of forbidden induced subgraphs, and present a linear-time algorithm to recognize them.

Another disjoint compression algorithm for odd cycle transversal

Given a graph G and an odd cycle transversal T, we describe an elegant O⁎(2|T|) algorithm for determining whether G has a smaller odd cycle transversal that is disjoint from T. We believe that our algorithm, based on a reduction to Vertex Cover, is conceptually simpler than the known algorithms for the problem and refines the understanding of the relationship between Odd Cycle Transversal and Vertex Cover.

Abstract 3574: Studies of castrate resistant 22Rv1 cells identifies AR regulated interrelated networks of transcription factors, co-regulators, chromatin, and nuclear scaffolding proteins.

Abstract Prostate cancer development, progression and acquisition of castrate resistance are reliant on the activity of the androgen receptor (AR), a transcription factor that governs the proliferation of prostate cancer cells. Multiple studies have focused on the principle components that are critical for cell cycle transversal and these studies have identified members of the E2F/RB family, c-Myc, cyclins and cyclin dependent kinase inhibitors as key proteins that are instrumental in prostate cell proliferation and development of castrate resistance. But the mechanism by which the AR dominates this process is unclear. To identify the link between the AR and cell cycle control we used 22Rv1 cells which are castrate resistant due to the expression of low molecular weight AR forms that are missing the ligand binding domain. However, a siRNA mediated decrease in AR expression induces a growth arrest confirming that proliferation is AR dependent. RNA-seq analysis of gene expression at castrate levels of androgen and following AR ablation identified transcripts that were AR dependent. This methodology reduced the number of AR regulated transcripts, by excluding transcripts regulated by super-physiological levels of androgens. The AR regulated transcripts included previously identified genes (PMEPA1, C1orf116, PCDH7, APP) including NFAT a gene shown to be regulated by a low molecular weight AR isoform. The analysis also revealed that interrelated networks of transcription factors and co-factors were AR transactivated or repressed. These included proteins such as the LBR, LMNB, TMPO and SATB1 that do not directly promote transcription, but rather link chromatin to the nuclear membrane or scaffolding, suggesting that the AR may modulate gene expression through epigenetics mechanisms. Network analysis showed connections between these proteins and cell cycle components including E2Fs and Myc where a number of the regulated transcripts are known E2F or Myc targets. In addition, this analysis identified a number of non-coding RNAs; some that have been previously described and some that have yet to be annotated. qPCR studies found that a number of the transcripts regulated in 22Rv1 cells are also AR regulated in LNCaP cells. Coupled with ChIP-seq data, the study characterizes the link between the AR and the cell cycle machinery and chromatin modulating proteins. Citation Format: Maria Mudryj, Stephen J. Libertini, Alan P. Lombard. Studies of castrate resistant 22Rv1 cells identifies AR regulated interrelated networks of transcription factors, co-regulators, chromatin, and nuclear scaffolding proteins. [abstract]. In: Proceedings of the 104th Annual Meeting of the American Association for Cancer Research; 2013 Apr 6-10; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2013;73(8 Suppl):Abstract nr 3574. doi:10.1158/1538-7445.AM2013-3574

On the anti-Kekulé number and odd cycle transversal of regular graphs

Let G be a graph with a perfect matching. The anti-Kekulé number is the smallest number of edges of G whose removal will result in a connected subgraph with no perfect matching. In this note, we show that, if G is a cyclically (r+1)-edge-connected r-regular graph (r≥3) of even order, then either the anti-Kekulé number of G is at least r+1, or G is not bipartite, and the smallest odd cycle transversal of G has at most r edges. For cubic graphs, the bound of the anti-Kekulé number is optimal. Our result generalizes previous results on anti-Kekulé numbers of fullerenes obtained by Kutnar et al. [K. Kutnar, J. Sedlar, D. Vukičević, On the anti-Kekulé number of leapfrog fullerenes, J. Math. Chem. 45 (2009) 431–441.] and Yang et al. [Q. Yang, D. Ye, H. Zhang, Y. Lin, On the anti-Kekulé number of fullerenes, MATCH Commun. Math. Comput. Chem. 67 (2) (2012) 281–288].

Obtaining a Bipartite Graph by Contracting Few Edges

The Bipartite Contraction problem takes as input an $n$-vertex graph $G$ and an integer $k$, and the task is to determine whether we can obtain a bipartite graph from $G$ by a sequence of at most $k$ edge contractions. We show that Bipartite Contraction is fixed-parameter tractable when parameterized by $k$. Despite a strong resemblance between Bipartite Contraction and the classical Odd Cycle Transversal (OCT) problem, the methods developed to tackle OCT do not seem to be directly applicable to Bipartite Contraction. To obtain our result, we combine several techniques and concepts that are central in parameterized complexity: iterative compression, irrelevant vertices, and important separators. To the best of our knowledge, this is the first time the irrelevant vertex technique and the concept of important separators are applied in unison. Furthermore, our algorithm may serve as a comprehensible example of the usage of the irrelevant vertex technique.

Cycle transversals in perfect graphs and cographs

A cycle transversal (or feedback vertex set) of a graph G is a subset T⊆V(G) such that T∩V(C)≠0̸ for every cycle C of G. This work considers the problem of finding special cycle transversals in perfect graphs and cographs. We prove that finding a minimum cycle transversal T in a perfect graph G is NP-hard, even for bipartite graphs with maximum degree four. Since G−T is acyclic, this result implies that finding a maximum acyclic induced subgraph of a perfect graph is also NP-hard. Other special properties of T are considered. A clique (stable, respectively) cycle transversal, or simply cct (sct, respectively) is a cycle transversal which is a clique (stable set, respectively). Recognizing graphs which admit a cct can be done in polynomial time; however, no structural characterization of such graphs is known, even for perfect graphs. We characterize cographs with cct in terms of forbidden induced subgraphs and describe their structure. This leads to linear time recognition of cographs with cct. We also prove that deciding whether a perfect graph admits an sct is NP-complete. We characterize cographs with sct in terms of forbidden induced subgraphs; this characterization also leads to linear time recognition.

Contracting Graphs to Paths and Trees

Vertex deletion and edge deletion problems play a central role in parameterized complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. The study of analogous edge contraction problems has so far been left largely unexplored from a parameterized perspective. We consider two basic problems of this type: Tree Contraction and Path Contraction. These two problems take as input an undirected graph G on n vertices and an integer k, and the task is to determine whether we can obtain a tree or a path, respectively, by a sequence of at most k edge contractions in G. For Tree Contraction, we present a randomized 4 k n O(1) time polynomial-space algorithm, as well as a deterministic 4.98 k n O(1) time algorithm, based on a variant of the color coding technique of Alon, Yuster and Zwick. We also present a deterministic 2 k+o(k)+n O(1) time algorithm for Path Contraction. Furthermore, we show that Path Contraction has a kernel with at most 5k+3 vertices, while Tree Contraction does not have a polynomial kernel unless NP ⊆ coNP/poly. We find the latter result surprising because of the connection between Tree Contraction and Feedback Vertex Set, which is known to have a kernel with 4k 2 vertices.

Kernelization hardness of connectivity problems in [formula omitted]-degenerate graphs

A graph is d-degenerate if every subgraph contains a vertex of degree at most d. For instance, planar graphs are 5-degenerate. Inspired by the recent work by Philip, Raman and Sikdar, who have shown the existence of a polynomial kernel for Dominating Set in d-degenerate graphs, we investigate the kernelization complexity of problems that include a connectivity requirement in this class of graphs.Our main contribution is the proof that Connected Dominating Set does not admit a polynomial kernel in d-degenerate graphs for d≥2 unless NP⊆coNP/poly, which is known to cause a collapse of the polynomial hierarchy to its third level. We prove this using a problem that originates from bioinformatics–Colourful Graph Motif–analysed and proved to be NP-hard by Fellows et al. This problem nicely encapsulates the hardness of the connectivity requirement in kernelization. Our technique also yields an alternative proof that, under the same complexity assumption, Steiner Tree does not admit a polynomial kernel. The original proof, via a reduction from Set Cover, is due to Dom, Lokshtanov and Saurabh.We extend our analysis by showing that, unless NP⊆coNP/poly, there do not exist polynomial kernels for Steiner Tree, Connected Feedback Vertex Set and Connected Odd Cycle Transversal in d-degenerate graphs for d≥2. On the other hand, we show a polynomial kernel for Connected Vertex Cover in graphs that do not contain the biclique Ki,j as a subgraph. As a d-degenerate graph cannot contain Kd+1,d+1 as a subgraph, the results holds also for graphs of bounded degeneracy.

Kernelization for cycle transversal problems

We present new kernelization results for two problems, s-cycle transversal and (≤s)-cycle transversal, when s is 4 or 5. We show that 4-cycle transversal and ≤4-cycle transversal admit 6k2 vertex kernels in general graphs. We then prove NP-completeness of s-cycle transversal and (≤s)-cycle transversal in planar graphs for s>3. We show the following linear vertex kernels in planar graphs—a 74k vertex kernel for 4-cycle transversal; a 32k vertex kernel for (≤4)-cycle transversal; a 266k vertex kernel for (≤5)-cycle transversal.

Odd Cycle Transversals and Independent Sets in Fullerene Graphs

A fullerene graph is a cubic bridgeless plane graph with all faces of size $5$ and $6$. We show that every fullerene graph on $n$ vertices can be made bipartite by deleting at most $\sqrt{12n/5}$ edges and has an independent set with at least $n/2-\sqrt{3n/5}$ vertices. Both bounds are sharp, and we characterize the extremal graphs. This proves conjectures of Došlić and Vukičević and of Daugherty. We deduce two further conjectures on the independence number of fullerene graphs, as well as a new upper bound on the smallest eigenvalue of a fullerene graph.

Local search: Is brute-force avoidable?

Many local search algorithms are based on searching in the k-exchange neighborhood. This is the set of solutions that can be obtained from the current solution by exchanging at most k elements. As a rule of thumb, the larger k is, the better are the chances of finding an improved solution. However, for inputs of size n, a naïve brute-force search of the k-exchange neighborhood requires nO(k) time, which is not practical even for very small values of k. We show that for several classes of sparse graphs, including planar graphs, graphs of bounded vertex degree and graphs excluding some fixed graph as a minor, an improved solution in the k-exchange neighborhood for many problems can be found much more efficiently. Our algorithms run in time O(τ(k)⋅nc), where τ is a function depending only on k and c is a constant independent of k and n. We demonstrate the applicability of this approach on a variety of problems including r-Center, Vertex Cover, Odd Cycle Transversal, Max-Cut, and Min-Bisection. In particular, on planar graphs, all our algorithms searching for a k-local improvement run in time O(2O(k)⋅n2), which is polynomial for k=O(logn). We complement these fixed-parameter tractable algorithms for k-local search with parameterized intractability results indicating that brute-force search is unavoidable in more general classes of graphs.

On the small cycle transversal of planar graphs

We consider the problem of finding a k -edge transversal set that intersects all (simple) cycles of length at most s in a planar graph, where s ≥ 3 is a constant. This problem, referred to as Small Cycle Transversal, is known to be NP-complete. We present a polynomial–time algorithm that computes a kernel of size 36 s 3 k for Small Cycle Transversal. In order to achieve this kernel, we extend the region decomposition technique of Alber et al. (2004) [1] by considering a unique region decomposition that is defined by shortest paths. Our kernel size is a significant improvement in terms of s over the kernel size obtained under the meta-kernelization framework by Bodlaender et al. (2009) [7].

Cycle transversals in bounded degree graphs

Graphs and Algorithms In this work we investigate the algorithmic complexity of computing a minimum C(k)-transversal, i.e., a subset of vertices that intersects all the chordless cycles with k vertices of the input graph, for a fixed k \textgreater= 3. For graphs of maximum degree at most three, we prove that this problem is polynomial-time solvable when k \textless= 4, and NP-hard otherwise. For graphs of maximum degree at most four, we prove that this problem is NP-hard for any fixed k \textgreater= 3. We also discuss polynomial-time approximation algorithms for computing C(3)-transversals in graphs of maximum degree at most four, based on a new decomposition theorem for such graphs that leads to useful reduction rules.

Cycle transversals in bounded degree graphs

In this work we consider the problem of finding a minimum Ck-transversal (a subset of vertices hitting all the induced chordless cycles with k vertices) in a graph with bounded maximum degree. In particular, we seek for dichotomy results as follows: for a fixed value of k, finding a minimum Ck-transversal is polynomial-time solvable if k ≤ p ,a nd NP -hard otherwise.

Finding odd cycle transversals

We present an O( mn) algorithm to determine whether a graph G with m edges and n vertices has an odd cycle transversal of order at most k, for any fixed k. We also obtain an algorithm that determines, in the same time, whether a graph has a half integral packing of odd cycles of weight k.

Cyclin A-CDK phosphorylates Sp1 and enhances Sp1-mediated transcription.

Cyclin A-mediated activation of cyclin-dependent kinases (CDKs) is essential for cell cycle transversal. Cyclin A activity is regulated on several levels and cyclin A elevation in a number of cancers suggests a role in tumorigenesis. In the present study, we used a modified DNA binding site selection and PCR amplification procedure to identify DNA binding proteins that are potential substrates of cyclin A-CDK. One of the sequences identified is the Sp1 transcription factor binding site. Co-immunoprecipitation experiments show that cyclin A and Sp1 can interact physically. In vitro and in vivo phosphorylation studies indicate that cyclin A-CDK complexes can phosphorylate Sp1. The phosphorylation site is located in the N-terminal region of the protein. Cells overexpressing cyclin A have elevated levels of Sp1 DNA binding activity, suggesting that cyclin A-CDK-mediated phosphorylation augments Sp1 DNA binding properties. In co-transfection studies, cyclin A expression stimulated transcription from an Sp1-regulated promoter. Mutation of the phosphorylation site abrogated cyclin A-CDK-dependent phosphorylation, augmentation of Sp1 transactivation function and DNA binding activity.

EGF induces cell cycle arrest of A431 human epidermoid carcinoma cells.

The human carcinoma cell line A431 is unusual in that physiologic concentrations of epidermal growth factor (EGF) inhibit proliferation. In the presence of 5-10 nM EGF proliferation of A431 cells is abruptly and markedly decreased compared to the untreated control cultures, with little loss of cell viability over a 4-day period. This study was initiated to examine how EGF affects the progression of A431 cells through the cell cycle. Flow cytometric analysis of DNA in EGF-treated cells reveals a marked change in the cell cycle distribution. The percentage of cells in late S/G2 increases and early S phase is nearly depleted. Since addition of the mitotic inhibitor vinblastine causes accumulation of cells in mitosis and prevents reentry of cells into G1, it is possible to distinguish between slow progression through G1 and G2 and blocks in those phases. When control cells, not treated with EGF, are exposed to vinblastine, the cells accumulate mitotic figures, as expected, and show progression into S, thus diminishing the number of cells in G1. In contrast, no mitotic figures are found among the EGF-treated cells in the presence or absence of vinblastine, and progression from G1 into S is not observed, as the number of cells in G1 remains constant. These results suggest that there are two EGF-induced blocks in cell cycle transversal; one is in late S and/or G2, blocking entry into mitosis, and the other is in G1, blocking entry into S phase. After 24 hours of EGF treatment, DNA synthesis is reduced to less than 10% compared to untreated controls as measured by the incorporation of [3H]thymidine or BrdU. In contrast, protein synthesis is inhibited by about twofold. Although inhibition of protein synthesis is less extensive, it occurs 6 hours prior to an equivalent inhibition of DNA synthesis. The rapid decrease in protein synthesis may result in the subsequent cell cycle arrest which occurs several hours later.