IA Scholar Query: A note on minimal directed graphs with given girth.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 04 Aug 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Substructures in Latin squares
https://scholar.archive.org/work/7s5oc53pq5g5djqancnbkzs7zm
We prove several results about substructures in Latin squares. First, we explain how to adapt our recent work on high-girth Steiner triple systems to the setting of Latin squares, resolving a conjecture of Linial that there exist Latin squares with arbitrarily high girth. As a consequence, we see that the number of order-n Latin squares with no intercalate (i.e., no 2×2 Latin subsquare) is at least (e^-9/4n-o(n))^n^2. Equivalently, ℙ[𝐍=0]≥ e^-n^2/4-o(n^2)=e^-(1+o(1))𝔼𝐍, where 𝐍 is the number of intercalates in a uniformly random order-n Latin square. In fact, extending recent work of Kwan, Sah, and Sawhney, we resolve the general large-deviation problem for intercalates in random Latin squares, up to constant factors in the exponent: for any constant 0<δ≤1 we have ℙ[𝐍≤(1-δ)𝔼𝐍]=exp(-Θ(n^2)) and for any constant δ>0 we have ℙ[𝐍≥(1+δ)𝔼𝐍]=exp(-Θ(n^4/3log n)). Finally, as an application of some new general tools for studying substructures in random Latin squares, we show that in almost all order-n Latin squares, the number of cuboctahedra (i.e., the number of pairs of possibly degenerate 2×2 submatrices with the same arrangement of symbols) is of order n^4, which is the minimum possible. As observed by Gowers and Long, this number can be interpreted as measuring "how associative" the quasigroup associated with the Latin square is.Matthew Kwan, Ashwin Sah, Mehtaab Sawhney, Michael Simkinwork_7s5oc53pq5g5djqancnbkzs7zmThu, 04 Aug 2022 00:00:00 GMTCommon Pairs of Graphs
https://scholar.archive.org/work/dgxxd3p2rvftfefcyso4xdqequ
A graph H is said to be common if the number of monochromatic labelled copies of H in a red/blue edge colouring of a large complete graph is asymptotically minimized by a random colouring with an equal proportion of each colour. We extend this notion to an asymmetric setting. That is, we define a pair (H_1,H_2) of graphs to be (p,1-p)-common if a particular linear combination of the density of H_1 in red and H_2 in blue is asymptotically minimized by a random colouring in which each edge is coloured red with probability p and blue with probability 1-p. We extend many of the results on common graphs to this asymmetric setting. In addition, we obtain several novel results for common pairs of graphs with no natural analogue in the symmetric setting. We also obtain new examples of common graphs in the classical sense and propose several open problems.Natalie Behague and Natasha Morrison and Jonathan A. Noelwork_dgxxd3p2rvftfefcyso4xdqequWed, 03 Aug 2022 00:00:00 GMTIncommensurable lattices in Baumslag-Solitar complexes
https://scholar.archive.org/work/lonvw6l4rvdllbapivdrhrw6ku
This paper concerns locally finite 2-complexes X_m,n which are combinatorial models for the Baumslag-Solitar groups BS(m,n). We show that, in many cases, the locally compact group Aut(X_m,n) contains incommensurable uniform lattices. The lattices we construct also admit isomorphic Cayley graphs and are finitely presented, torsion-free, and coherent.Max Foresterwork_lonvw6l4rvdllbapivdrhrw6kuFri, 29 Jul 2022 00:00:00 GMTVarieties of Nodal surfaces, coding theory and Discriminants of cubic hypersurfaces. Part 1: Generalities and nodal K3 surfaces. Part 2: Cubic Hypersurfaces, associated discriminants. Part 3: Nodal quintics. Part 4: Nodal sextics
https://scholar.archive.org/work/x3qw3eeaxfc7vda5csvzhnt7ii
In this paper we exploit two binary codes associated to a projective nodal surface (the strict code K and, for even degree d, the extended code K' ) to investigate the 'Nodal Severi varieties F(d, n) of nodal surfaces in P^3 of degree d and with n nodes, and their incidence hierarchy, relating partial smoothings to code shortenings. Our first main result solves a question which dates back over 100 years: the irreducible components of F(4, n) are in bijection with the isomorphism classes of their extended codes K', and these are exactly all the 34 possible shortenings of the extended Kummer code K' , and a component is in the closure of another if and only if the code of the latter is a shortening of the code of the former. We then extend this result to nodal K3 surfaces of all other degrees d = 2h in an entirely analogous way, showing that some unexpected families occur. For surfaces of degree d=5 in P^3 we determine (with one possible exception) all the possible codes K of nodal quintics, and for several cases of K, we show the irreducibility of the corresponding open set of F(5, n), for instance we show the irreducibility of the family of Togliatti quintic surfaces. In the fourth part we show that a 'Togliatti-like' description holds for surfaces of degree 6 with the maximum number of nodes= 65: they are discriminants of cubic hypersurfaces in P^6 with 31 (respectively 32) nodes, and we have an irreducible 18-dimensional family of them. For degree d=6, our main result is based on some novel auxiliary results: 1) the study of the half-even sets of nodes on sextic surfaces, 2) the investigation of discriminants of cubic hypersurfaces X, 3) the computer assisted proof that, for n = 65, both codes K, K' are uniquely determined, 4) the description of these codes, relating the geometry of the Barth sextic with the Doro-Hall graph.Fabrizio Catanese, Yonghwa Cho, Stephen Coughlan, Davide Frapporti, Alessandro Verra, Michael Kiermaierwork_x3qw3eeaxfc7vda5csvzhnt7iiFri, 29 Jul 2022 00:00:00 GMTExtremal and monotone behaviour of the Sudoku number and related critical set parameters
https://scholar.archive.org/work/yxx6bqlwtvfxtmklmoshjdzxu4
The Sudoku number has been defined under various names, indicating it is a natural concept. There are four variants of this parameter, that can be related to the maximum and minimum size of a critical set in a graph colouring problem. For each of these four related parameters, we present some simple characterizations of the graphs attaining the maximum possible values. As a main result, we answer a question by Cooper and Kirkpatrick, showing that there is monotone behaviour in the number of colours for only two of the four parameters. We investigate the monotone behaviour for the subgraph-order as well. For Latin squares and the Sudoku, we solve some variants for hypergraph colouring.Stijn Cambiework_yxx6bqlwtvfxtmklmoshjdzxu4Thu, 28 Jul 2022 00:00:00 GMTRepresentation of short distances in structurally sparse graphs
https://scholar.archive.org/work/gc6fdxey55bgvj7sjtuhih2v5e
A partial orientation H⃗ of a graph G is a weak r-guidance system if for any two vertices at distance at most r in G, there exists a shortest path P between them such that H⃗ directs all but one edge in P towards this edge. In case H⃗ has bounded maximum outdegree, this gives an efficient representation of shortest paths of length at most r in G. We show that graphs from many natural graph classes admit such weak guidance systems, and study the algorithmic aspects of this notion.Zdeněk Dvořákwork_gc6fdxey55bgvj7sjtuhih2v5eThu, 28 Jul 2022 00:00:00 GMTA SAT Attack on Rota's Basis Conjecture
https://scholar.archive.org/work/s7kunpczlna3vatluxqlucu36e
The SAT modulo Symmetries (SMS) is a recently introduced framework for dynamic symmetry breaking in SAT instances. It combines a CDCL SAT solver with an external lexicographic minimality checking algorithm. We extend SMS from graphs to matroids and use it to progress on Rota's Basis Conjecture (1989), which states that one can always decompose a collection of r disjoint bases of a rank r matroid into r disjoint rainbow bases. Through SMS, we establish that the conjecture holds for all matroids of rank 4 and certain special cases of matroids of rank 5. Furthermore, we extend SMS with the facility to produce DRAT proofs. External tools can then be used to verify the validity of additional axioms produced by the lexicographic minimality check. As a byproduct, we have utilized our framework to enumerate matroids modulo isomorphism and to support the investigation of various other problems on matroids.Markus Kirchweger, Manfred Scheucher, Stefan Szeider, Kuldeep S. Meel, Ofer Strichmanwork_s7kunpczlna3vatluxqlucu36eThu, 28 Jul 2022 00:00:00 GMTA degree preserving delta wye transformation with applications to 6-regular graphs and Feynman periods
https://scholar.archive.org/work/bygowdvleraw7kdjvaot4isj24
We investigate a degree preserving variant of the Δ-Y transformation which replaces a triangle with a new 6-valent vertex which has double edges to the vertices that had been in the triangle. This operation is relevant for understanding scalar Feynman integrals in 6 dimensions. We study the structure of equivalence classes under this operation and its inverse, with particular attention to when the equivalence classes are finite, when they contain simple 6-regular graphs, and when they contain doubled 3-regular graphs. The last of these, in particular, is relevant for the Feynman integral calculations and we make some observations linking the structure of these classes to the Feynman periods. Furthermore, we investigate properties of minimal graphs in these equivalence classes.Shannon Jeffries, Karen Yeatswork_bygowdvleraw7kdjvaot4isj24Wed, 27 Jul 2022 00:00:00 GMTTwo-geodesic transitive graphs of order p^n with n≤3
https://scholar.archive.org/work/nuwzcxtfijgelahwteweqskyaq
A vertex triple (u,v,w) of a graph is called a 2-geodesic if v is adjacent to both u and w and u is not adjacent to w. A graph is said to be 2-geodesic transitive if its automorphism group is transitive on the set of 2-geodesics. In this paper, a complete classification of 2-geodesic transitive graphs of order p^n is given for each prime p and n≤ 3. It turns out that all such graphs consist of three small graphs: the complete bipartite graph K_4,4 of order 8, the Schläfli graph of order 27 and its complement, and fourteen infinite families: the cycles C_p, C_p^2 and C_p^3, the complete graphs K_p, K_p^2 and K_p^3, the complete multipartite graphs K_p[p], K_p[p^2] and K_p^2[p], the Hamming graph H(2,p) and its complement, the Hamming graph H(3,p), and two infinite families of normal Cayley graphs on extraspecial group of order p^3 and exponent p.Jun-Jie Huang, Yan-Quan Feng, Jin-Xin Zhou, Fu-Gang Yinwork_nuwzcxtfijgelahwteweqskyaqWed, 27 Jul 2022 00:00:00 GMTCombinatorial Gray codes-an updated survey
https://scholar.archive.org/work/zryp7sxkrbczrguasg4ugmgfee
A combinatorial Gray code for a class of objects is a listing that contains each object from the class exactly once such that any two consecutive objects in the list differ only by a 'small change'. Such listings are known for many different combinatorial objects, including bitstrings, combinations, permutations, partitions, triangulations, but also for objects defined with respect to a fixed graph, such as spanning trees, perfect matchings or vertex colorings. This survey provides a comprehensive picture of the state-of-the-art of the research on combinatorial Gray codes. In particular, it gives an update on Savage's influential survey [C. D. Savage. A survey of combinatorial Gray codes. SIAM Rev., 39(4):605-629, 1997.], incorporating many more recent developments. We also elaborate on the connections to closely related problems in graph theory, algebra, order theory, geometry and algorithms, which embeds this research area into a broader context. Lastly, we collect and propose a number of challenging research problems, thus stimulating new research endeavors.Torsten Mützework_zryp7sxkrbczrguasg4ugmgfeeTue, 26 Jul 2022 00:00:00 GMTReciprocal SOX2 regulation by SMAD1-SMAD3 is critical for anoikis resistance and metastasis in cancer
https://scholar.archive.org/work/gz3evgzdufdhhl2x64dh36ffgi
Growth factors in tumor environments are regulators of cell survival and metastasis. Here, we reveal the dichotomy between TGF-β superfamily growth factors BMP and TGF-β/activin and their downstream SMAD effectors. Gene expression profiling uncovers SOX2 as a key contextual signaling node regulated in an opposing manner by BMP2, -4, and -9 and TGF-β and activin A to impact anchorage-independent cell survival. We find that SOX2 is repressed by BMPs, leading to a reduction in intraperitoneal tumor burden and improved survival of tumor-bearing mice. Repression of SOX2 is driven by SMAD1-dependent histone H3K27me3 recruitment and DNA methylation at SOX2's promoter. Conversely, TGF-β, which is elevated in patient ascites, and activin A can promote SOX2 expression and anchorage-independent survival by SMAD3-dependent histone H3K4me3 recruitment. Our findings identify SOX2 as a contextual and contrastingly regulated node downstream of TGF-β members controlling anchorage-independent survival and metastasis in ovarian cancers.Zainab Shonibare, Mehri Monavarian, Kathleen O'Connell, Diego Altomare, Abigail Shelton, Shubham Mehta, Renata Jaskula-Sztul, Rebecca Phaeton, Mark D. Starr, Regina Whitaker, Andrew Berchuck, Andrew B. Nixon, Rebecca C. Arend, Nam Y. Lee, C. Ryan Miller, Nadine Hempel, Karthikeyan Mythreyework_gz3evgzdufdhhl2x64dh36ffgiTue, 26 Jul 2022 00:00:00 GMTCommon graphs with arbitrary chromatic number
https://scholar.archive.org/work/34q5rwajmrh4lkxsnnhuntgcqe
Ramsey's Theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erdos conjectured that the random 2-edge-coloring minimizes the number of monochromatic copies of K_k, and the conjecture was extended by Burr and Rosta to all graphs. In the late 1980s, the conjectures were disproved by Thomason and Sidorenko, respectively. A classification of graphs whose number of monochromatic copies is minimized by the random 2-edge-coloring, which are referred to as common graphs, remains a challenging open problem. If Sidorenko's Conjecture, one of the most significant open problems in extremal graph theory, is true, then every 2-chromatic graph is common, and in fact, no 2-chromatic common graph unsettled for Sidorenko's Conjecture is known. While examples of 3-chromatic common graphs were known for a long time, the existence of a 4-chromatic common graph was open until 2012, and no common graph with a larger chromatic number is known. We construct connected k-chromatic common graphs for every k. This answers a question posed by Hatami, Hladky, Kral, Norine and Razborov [Combin. Probab. Comput. 21 (2012), 734-742], and a problem listed by Conlon, Fox and Sudakov [London Math. Soc. Lecture Note Ser. 424 (2015), 49-118, Problem 2.28]. This also answers in a stronger form the question raised by Jagger, Stovicek and Thomason [Combinatorica 16, (1996), 123-131] whether there exists a common graph with chromatic number at least four.Daniel Kral and Jan Volec and Fan Weiwork_34q5rwajmrh4lkxsnnhuntgcqeFri, 22 Jul 2022 00:00:00 GMTA faster algorithm for Vertex Cover parameterized by solution size
https://scholar.archive.org/work/2jhbwz5a4fbkljwpq5ackaf6h4
We describe a new algorithm for vertex cover with runtime O^*(1.25298^k), where k is the size of the desired solution and O^* hides polynomial factors in the input size. This improves over previous runtime of O^*(1.2738^k) due to Chen, Kanj, Xia (2010) standing for more than a decade. The key to our algorithm is to use a potential function which simultaneously tracks k as well as the optimal value λ of the vertex cover LP relaxation. This approach also allows us to make use of prior algorithms for Maximum Independent Set in bounded-degree graphs and Above-Guarantee Vertex Cover. The main step in the algorithm is to branch on high-degree vertices, while ensuring that both k and μ = k - λ are decreased at each step. There can be local obstructions in the graph that prevent μ from decreasing in this process; we develop a number of novel branching steps to handle these situations.David G. Harris, N. S. Narayanaswamywork_2jhbwz5a4fbkljwpq5ackaf6h4Wed, 20 Jul 2022 00:00:00 GMTNode and Edge Averaged Complexities of Local Graph Problems
https://scholar.archive.org/work/a5ejim3cazep7fy4yionno43hi
We continue the recently started line of work on the distributed node-averaged complexity of distributed graph algorithms. The node-averaged complexity of a distributed algorithm running on a graph 𝐺 = (𝑉 , 𝐸) is the average over the times at which the nodes 𝑉 of 𝐺 finish their computation and commit to their outputs. We study the node-averaged complexity for some of the central distributed symmetry breaking problems. As our main result, we show that the randomized node-averaged complexity of computing a maximal independent set (MIS) is at least Ω(min{log Δ/log log Δ, √︁ log 𝑛/log log 𝑛}) in 𝑛-node graphs of maximum degree Δ. This bound is obtained by a novel adaptation of the well-known lower bound of Kuhn, Moscibroda, and Wattenhofer [JACM'16]. As a side result, we obtain that the worstcase randomized round complexity for computing an MIS in trees is also Ω(min{log Δ/log log Δ, √︁ log 𝑛/log log 𝑛})-this essentially answers open problem 11.15 in the book of Barenboim and Elkin and resolves the complexity of MIS on trees up to an 𝑂 ( √︁ log log 𝑛) factor. We also show that, perhaps surprisingly, a minimal relaxation of MIS, which is the same as (2, 1)-ruling set, to the (2, 2)-ruling set problem drops the randomized node-averaged complexity to 𝑂 (1). For the problem of computing a maximal matching, we show that while the randomized node-averaged complexity is at least Ω(min{log Δ/log log Δ, √︁ log 𝑛/log log 𝑛}), the randomized edgeaveraged complexity is 𝑂 (1). Further, we show that the deterministic edge-averaged complexity of maximal matching is 𝑂 (log 2 Δ + log * 𝑛) and the deterministic node-averaged complexity of maximal matching is 𝑂 (log 3 Δ + log * 𝑛). Finally, we consider the problem of computing a sinkless orientation of a graph. The deterministic worst-case complexity of the problem is known to be Θ(log 𝑛), even on bounded-degree graphs. We show that the problem can be solved deterministically with node-averaged complexity 𝑂 (log * 𝑛), while keeping the worst-case complexity in 𝑂 (log 𝑛).Alkida Balliu, Mohsen Ghaffari, Fabian Kuhn, Dennis Olivettiwork_a5ejim3cazep7fy4yionno43hiWed, 20 Jul 2022 00:00:00 GMTThe Landscape of Distributed Complexities on Trees and Beyond
https://scholar.archive.org/work/6mfz5bl6pne2dnycb52656jvye
We study the local complexity landscape of locally checkable labeling (LCL) problems on constant-degree graphs with a focus on complexities below log * 𝑛. Our contribution is threefold: (1) Our main contribution is that we complete the classification of the complexity landscape of LCL problems on trees in the LOCAL model, by proving that every LCL problem with local complexity 𝑜 (log * 𝑛) has actually complexity 𝑂 (1). This result improves upon the previous speedup result from 𝑜 (log log * 𝑛) to 𝑂 (1) by [Chang, Pettie, FOCS 2017]. (2) In the related LCA and VOLUME models [Alon, Rubinfeld, Vardi, Xie, SODA 2012, Rubinfeld, Tamir, Vardi, Xie, 2011, Rosenbaum, Suomela, PODC 2020], we prove the same speedup from 𝑜 (log * 𝑛) to 𝑂 (1) for all constant-degree graphs. (3) Similarly, we complete the classification of the LOCAL complexity landscape of oriented 𝑑-dimensional grids by proving that any LCL problem with local complexity 𝑜 (log * 𝑛) has actually complexity 𝑂 (1). This improves upon the previous speed-up from 𝑜 ( 𝑑 √︁ log * 𝑛) by Suomela, explained in [Chang, Pettie, FOCS 2017]. CCS CONCEPTS • Theory of computation → Distributed algorithms.Christoph Grunau, Václav Rozhoň, Sebastian Brandtwork_6mfz5bl6pne2dnycb52656jvyeWed, 20 Jul 2022 00:00:00 GMTSurface and borehole geophysical analysis of structures within the Callide Basin, eastern Central Queensland
https://scholar.archive.org/work/dlfqfjg3tvgthap2kwueadza7a
Traditional geophysical techniques, such as electrical, magnetic, seismic and gamma spectroscopic methods, have been deployed across the Callide Basin, Eastern Central Queensland, intent on delineating basin -wide structures. Further, innovative surface and borehole geophysical techniques have been applied for coal mine -scale exploration and production with the intention of reducing global geological ambiguity and optimising exploration resources at Callide Coalfields. A very low frequency electromagnetic surface impedance mapping method, the SIROLOG downhole technique, acoustic scanning, electromagnetic tomography and full wave -form sonic borehole logging have been trialed for geological hazard and mine design applications at Callide Coalfields as the precursor to their wider application and acceptance in the Australian coal industry. In this thesis, the theoretical basis for these techniques is provided. However, more importantly, the case studies presented demonstrate the role that these geophysical techniques have played in identifying geological structures critical to mining. Reverse faults that daylight in highwalls and intrusions constitute geological hazards that affect safety, costs and scheduling in mining operations. Identification of the limit of oxidation of coal seams (coal subcrop) is critical in mine design. During the course of this thesis, the application of geophysical techniques resulted in: a) a major structure (the "Trap Gully Monocline") being redefined from its original interpretation as a normal fault to a monocline that is stress -relieved by minor scale thrust faulting; b) two previously unidentified intrusions (the Kilburnie "Homestead" plug and The Hut "Crater" plug) that impinge on mining have been discovered; c) the delineation of two coal subcrop lines has resulted in the discovery of an additional 1.5 million tonnes of coal reserve at Boundary Hill mine and the successful redesign of mining strips at The Hut Central Valley and Eastern Hillside brownfield sites; and d) [...]Wesley James Foi Nicholswork_dlfqfjg3tvgthap2kwueadza7aTue, 19 Jul 2022 00:00:00 GMTSeti and the media: Improving science communication
https://scholar.archive.org/work/wiw45ppx2zgh5pwzbrjdqwuyaq
From its beginnings, the scientific Search for Extraterrestrial Intelligence has faced the 'giggle factor' - that all it amounted to was 'looking for Little Green Men'. Yet SETI has gained credibility as well as recognition that the endeavour is very much part of the rapidly emerging science of astrobiology. SETI is also unusual among areas of science in that almost from the beginning, researchers have considered the social and cultural implications of the experiment. Over the past 15 years, the SETI Institute in Mountain View, California, the largest organisation among a group of independent international efforts, has developed formal education curricula reflective of its research, which continues today. The Institute is also engaged in public outreach in an effort to improve the public understanding of SETI and SETI-related science. In particular, SETI has encouraged mass media attention through a variety of initiatives. This thesis will view science communication through the experiences of SETI - and mostly the SETI Institute. This - probably unique - approach will explore relevant elements of SETI and science communication to show that the current perspective of promulgating the public understanding of science via the mass media may be flawed and worthy of further investigation.Carol Ann Oliverwork_wiw45ppx2zgh5pwzbrjdqwuyaqTue, 19 Jul 2022 00:00:00 GMTThe immersion-minimal infinitely edge-connected graph
https://scholar.archive.org/work/zpyrveiuiffuzau32ovbiusa6i
We show that there is a unique immersion-minimal infinitely edge-connected graph: every such graph contains the halved Farey graph, which is itself infinitely edge-connected, as an immersion minor. By contrast, any minimal list of infinitely edge-connected graphs represented in all such graphs as topological minors must be uncountable.Paul Knappe, Jan Kurkofkawork_zpyrveiuiffuzau32ovbiusa6iMon, 18 Jul 2022 00:00:00 GMTProduct structure of graph classes with bounded treewidth
https://scholar.archive.org/work/j4bhjoyxnnevtnltqxhsrgt6hq
We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the "underlying treewidth" of a graph class 𝒢 to be the minimum non-negative integer c such that, for some function f, for every graph G ∈𝒢 there is a graph H with tw(H) ≤ c such that G is isomorphic to a subgraph of H ⊠ K_f(tw(G)). We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth 3; the class of K_s,t-minor-free graphs has underlying treewidth s (for t ≥max{s,3}); and the class of K_t-minor-free graphs has underlying treewidth t-2. In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no H subgraph has bounded underlying treewidth if and only if every component of H is a subdivided star, and that the class of graphs with no induced H subgraph has bounded underlying treewidth if and only if every component of H is a star.Rutger Campbell and Katie Clinch and Marc Distel and J. Pascal Gollin and Kevin Hendrey and Robert Hickingbotham and Tony Huynh and Freddie Illingworth and Youri Tamitegama and Jane Tan and David R. Woodwork_j4bhjoyxnnevtnltqxhsrgt6hqMon, 18 Jul 2022 00:00:00 GMTVertex Deletion Parameterized by Elimination Distance and Even Less
https://scholar.archive.org/work/sdzr3cd7lrdmnjd5v32mha6lde
We study the parameterized complexity of various classic vertex-deletion problems such as Odd cycle transversal, Vertex planarization, and Chordal vertex deletion under hybrid parameterizations. Existing FPT algorithms for these problems either focus on the parameterization by solution size, detecting solutions of size k in time f(k) · n^O(1), or width parameterizations, finding arbitrarily large optimal solutions in time f(w) · n^O(1) for some width measure w like treewidth. We unify these lines of research by presenting FPT algorithms for parameterizations that can simultaneously be arbitrarily much smaller than the solution size and the treewidth. We consider two classes of parameterizations which are relaxations of either treedepth of treewidth. They are related to graph decompositions in which subgraphs that belong to a target class H (e.g., bipartite or planar) are considered simple. First, we present a framework for computing approximately optimal decompositions for miscellaneous classes H. Namely, if the cost of an optimal decomposition is k, we show how to find a decomposition of cost k^O(1) in time f(k) · n^O(1). This is applicable to any graph class H for which the corresponding vertex-deletion problem admits a constant-factor approximation algorithm or an FPT algorithm paramaterized by the solution size. Secondly, we exploit the constructed decompositions for solving vertex-deletion problems by extending ideas from algorithms using iterative compression and the finite state property. For the three mentioned vertex-deletion problems, and all problems which can be formulated as hitting a finite set of connected forbidden (a) minors or (b) (induced) subgraphs, we obtain FPT algorithms with respect to both studied parameterizations.Bart M. P. Jansen, Jari J. H. de Kroon, Michał Włodarczykwork_sdzr3cd7lrdmnjd5v32mha6ldeMon, 18 Jul 2022 00:00:00 GMT