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Avoidable paths in graphs
[article]

2019
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arXiv
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pre-print

We prove a recent conjecture of Beisegel et al. that for every positive integer k, every graph containing an induced P_k also contains an avoidable P_k. Avoidability generalises the notion of simpliciality best known in the context of chordal graphs. The conjecture was only established for k in 1,2 (Ohtsuki et al. 1976, and Beisegel et al. 2019, respectively). Our result also implies a result of Chvátal et al. 2002, which assumed cycle restrictions. We provide a constructive and elementary

arXiv:1908.03788v1
fatcat:vyldpisxcjgmhppfchicn7svwq
## more »

... , relying on a single trick regarding the induction hypothesis. In the line of previous works, we discuss conditions for multiple avoidable paths to exist.##
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On Vizing's edge colouring question
[article]

2021
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arXiv
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pre-print

Soon after his 1964 seminal paper on edge colouring, Vizing asked the following question: can an optimal edge colouring be reached from any given proper edge colouring through a series of Kempe changes? We answer this question in the affirmative for triangle-free graphs.

arXiv:2107.07900v1
fatcat:hvjcg6srrbc6pkvvtftnzlnpeu
##
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Close relatives (of Feedback Vertex Set), revisited
[article]

2021
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arXiv
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pre-print

At IPEC 2020, Bergougnoux, Bonnet, Brettell, and Kwon showed that a number of problems related to the classic Feedback Vertex Set (FVS) problem do not admit a 2^o(k log k)· n^𝒪(1)-time algorithm on graphs of treewidth at most k, assuming the Exponential Time Hypothesis. This contrasts with the 3^k· k^𝒪(1)· n-time algorithm for FVS using the Cut Count technique. During their live talk at IPEC 2020, Bergougnoux et al. posed a number of open questions, which we answer in this work. - Subset Even

arXiv:2106.16015v1
fatcat:7zb56fhkfrfa3nl4bngr2phxwe
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... cle Transversal, Subset Odd Cycle Transversal, Subset Feedback Vertex Set can be solved in time 2^𝒪(k log k)· n in graphs of treewidth at most k. This matches a lower bound for Even Cycle Transversal of Bergougnoux et al. and improves the polynomial factor in some of their upper bounds. - Subset Feedback Vertex Set and Node Multiway Cut can be solved in time 2^𝒪(k log k)· n, if the input graph is given as a clique-width expression of size n and width k. - Odd Cycle Transversal can be solved in time 4^k · k^𝒪(1)· n if the input graph is given as a clique-width expression of size n and width k. Furthermore, the existence of a constant ε > 0 and an algorithm performing this task in time (4-ε)^k · n^𝒪(1) would contradict the Strong Exponential Time Hypothesis.##
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Avoidable Paths in Graphs

2020
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Electronic Journal of Combinatorics
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We prove a recent conjecture of Beisegel et al. that for every positive integer $k$, every graph containing an induced $P_k$ also contains an avoidable $P_k$. Avoidability generalises the notion of simpliciality best known in the context of chordal graphs. The conjecture was only established for $k \in \{1,2\}$ (Ohtsuki et al. 1976, and Beisegel et al. 2019, respectively). Our result also implies a result of Chvátal et al. 2002, which assumed cycle restrictions. We provide a constructive and

doi:10.37236/9030
fatcat:dova7wywhretnjonxjg735oqxe
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... mentary proof, relying on a single trick regarding the induction hypothesis. In the line of previous works, we discuss conditions for multiple avoidable paths to exist.##
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On objects dual to tree-cut decompositions
[article]

2022
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arXiv
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pre-print

Tree-cut width is a graph parameter introduced by Wollan that is an analogue of treewidth for the immersion order on graphs in the following sense: the tree-cut width of a graph is functionally equivalent to the largest size of a wall that can be found in it as an immersion. In this work we propose a variant of the definition of tree-cut width that is functionally equivalent to the original one, but for which we can state and prove a tight duality theorem relating it to naturally defined dual

arXiv:2103.14667v2
fatcat:lkil7zlcjza5fpvct6jjfnbijy
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... jects: appropriately defined brambles and tangles. Using this result we also propose a game characterization of tree-cut width.##
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Enumerating minimal dominating sets in the (in)comparability graphs of bounded dimension posets
[article]

2020
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arXiv
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pre-print

Enumerating minimal transversals in a hypergraph is a notoriously hard problem. It can be reduced to enumerating minimal dominating sets in a graph, in fact even to enumerating minimal dominating sets in an incomparability graph. We provide an output-polynomial time algorithm for incomparability graphs whose underlying posets have bounded dimension. Through a different proof technique, we also provide an output-polynomial algorithm for their complements, i.e., for comparability graphs of

arXiv:2004.07214v1
fatcat:5atat6z2hrbzxci2iv7kbumrfa
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... dimension posets. Our algorithm for incomparability graphs is based on flashlight search and relies on the geometrical representation of incomparability graphs with bounded dimension, as given by Golumbic et al. in 1983. It runs with polynomial delay and only needs polynomial space. Our algorithm for comparability graphs is based on the flipping method introduced by Golovach et al. in 2015. It performs in incremental-polynomial time and requires exponential space. In addition, we show how to improve the flipping method so that it requires only polynomial space. Since the flipping method is a key tool for the best known algorithms enumerating minimal dominating sets in a number of graph classes, this yields direct improvements on the state of the art.##
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On the dualization in distributive lattices and related problems
[article]

2020
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arXiv
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pre-print

In this paper, we study the dualization in distributive lattices, a generalization of the well-known hypergraph dualization problem. We in particular propose equivalent formulations of the problem in terms of graphs, hypergraphs, and posets. It is known that hypergraph dualization amounts to generate all minimal transversals of a hypergraph, or all minimal dominating sets of a graph. In this new framework, a poset on vertices is given together with the input (hyper)graph, and minimal "ideal

arXiv:1902.07004v2
fatcat:w7xlm72dqvbe3e7u3ihbuuoem4
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... tions" are to be generated. This in particular allows us to study the complexity of the problem under various combined restrictions on graph classes and poset types, including bipartite, split, and co-bipartite graphs, and variants of neighborhood inclusion posets. We for example show that while the enumeration of minimal dominating sets is possible with linear delay in split graphs, the problem, within the same class, gets as hard as for general graphs when generalized to this framework. More surprisingly, this result holds even when the poset is only comparing vertices of included neighborhoods in the graph. If both the poset and the graph class are sufficiently restricted, we show that the dualization is tractable relying on existing algorithms from the literature.##
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Translating between the representations of a ranked convex geometry
[article]

2021
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arXiv
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pre-print

It is well known that every closure system can be represented by an implicational base, or by the set of its meet-irreducible elements. In Horn logic, these are respectively known as the Horn expressions and the characteristic models. In this paper, we consider the problem of translating between the two representations in acyclic convex geometries. Quite surprisingly, we show that the problem in this context is already harder than the dualization in distributive lattices, a generalization of

arXiv:1907.09433v2
fatcat:zshldvesr5d6hjoyvdf4dpxb7m
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... well-known hypergraph dualization problem for which the existence of an output quasi-polynomial time algorithm is open. In light of this result, we consider a proper subclass of acyclic convex geometries, namely ranked convex geometries, as those that admit a ranked implicational base analogous to that of ranked posets. For this class, we provide output quasi-polynomial time algorithms based on hypergraph dualization for translating between the two representations. This improves the understanding of a long-standing open problem.##
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Revisiting a Theorem by Folkman on Graph Colouring

2020
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Electronic Journal of Combinatorics
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We give a short proof of the following theorem due to Jon H. Folkman (1969): The chromatic number of any graph is at most $2$ plus the maximum over all subgraphs of the difference between the number of vertices and twice the independence number.

doi:10.37236/8899
fatcat:ot3f2veitzbovjka2r7iz3kgym
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Enumerating Minimal Dominating Sets in Triangle-Free Graphs

2019
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Symposium on Theoretical Aspects of Computer Science
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It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this paper we prove that this is the case in triangle-free graphs. This answers a question of Kanté et al. Additionally, we show that deciding if a set of vertices of a bipartite graph can be completed into a minimal dominating set is a NP-complete problem. ACM Subject Classification Mathematics of computing → Graph algorithms; Theory of computation → Design and

doi:10.4230/lipics.stacs.2019.16
dblp:conf/stacs/BonamyDHR19
fatcat:2qvswqgw3fez5l3fs44c4urql4
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... sis of algorithms Acknowledgements The authors wish to thank Paul Ouvrard for extensive discussions on the topic of this paper. We also gratefully acknowledge support from Nicolas Bonichon and the Simon family for the organization of the 3 rd Pessac Graph Workshop, where this research was done. Last but not least, we thank Peppie for her unwavering support during the work sessions. The objects we wish to enumerate in this paper are the (inclusion-wise) minimal dominating sets of a given graph. In general, the number of these objects may grow exponentially with the order n of the input graph. Therefore, in stark contrast to decision or optimization problems, looking for a running time polynomially bounded by n is not a reasonable, let alone meaningful, efficiency criterion. Rather, we aim here for algorithms whose running time is polynomially bounded by the size of both the input and output data, called output-polynomial algorithms. Because dominating sets are among the most studied objects in graph theory and algorithms, their enumeration (and counting) have attracted an increasing attention over the past 10 years. The problem of enumerating minimal dominating sets (hereafter referred to as Dom-Enum) has a notable feature: it is equivalent to the extensively studied hypergraph problem Trans-Enum. In Trans-Enum, one is given a hypergraph H (i.e. a collection of sets, called hyperedges) and is asked to enumerate all the minimal transversals of H (i.e. the inclusion-minimal sets of elements that meet every hyperedge). It is not hard to see that Dom-Enum is a particular case of Trans-Enum: the minimal dominating sets of a graph G are exactly the minimal transversals of the hypergraph of closed neighborhoods of G. Conversely, Kanté, Limouzy, Mary, and Nourine proved that every instance of Trans-Enum can be reduced to a co-bipartite 1 instance of Dom-Enum [17] . Currently, the best output-sensitive algorithm for Trans-Enum is due to Fredman and Khachiyan and runs in quasi-polynomial time [9] . It is a long-standing open problem whether this complexity bound can be improved (see for instance the surveys [6, 8] ). Therefore, the equivalence between the two problems is an additional motivation to study Dom-Enum, with the hope that techniques from graph theory will be used to obtain new results on the Trans-Enum problem. So far, output-polynomial algorithms have been obtained for Dom-Enum in several classes of graphs, including planar graphs and degenerate graphs [7], classes of graphs of bounded tree-width, clique-width [4], or mim-width [10], path graphs and line graphs [16], interval graphs and permutation graphs [18], split graphs [19], graphs of girth at least 7 [12], chordal graphs [19], and chordal bipartite graphs [11]. A succinct survey of results on Dom-Enum can be found in [20] . The authors of [19] state as an open problem the question to design an output-polynomial algorithm for bipartite graphs (the problem also appeared in [20, 11] ). We address this problem with the following result. S TA C S 2 0 1 9##
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Revisiting a theorem by Folkman on graph colouring
[article]

2019
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arXiv
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pre-print

We give a short proof of the following theorem due to Jon H. Folkman (1969): The chromatic number of any graph is at most 2 plus the maximum over all subgraphs of the difference between half the number of vertices and the independence number.

arXiv:1907.11429v1
fatcat:p22hs6m57jfcxi3uzjhv46ew2a
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Dualization in lattices given by implicational bases
[article]

2020
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arXiv
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pre-print

It was recently proved that the dualization in lattices given by implicational bases is impossible in output-polynomial time unless P=NP. In this paper, we~show that this result holds even when the premises in the implicational base are of size at most two. Then we show using hypergraph dualization that the problem can be solved in output quasi-polynomial time whenever the implicational base has bounded independent-width, defined as the size of a maximum set of implications having independent

arXiv:1901.07503v5
fatcat:wm7iya77hjdm5mrc4nwd36d4e4
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... nclusions. Lattices that share this property include distributive lattices coded by the ideals of an interval order, when both the independent-width and the size of the premises equal one.##
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Connected greedy colourings of perfect graphs and other classes: the good, the bad and the ugly
[article]

2021
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arXiv
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pre-print

The Grundy number of a graph is the maximum number of colours used by the "First-Fit" greedy colouring algorithm over all vertex orderings. Given a vertex ordering σ= v_1,...,v_n, the "First-Fit" greedy colouring algorithm colours the vertices in the order of σ by assigning to each vertex the smallest colour unused in its neighbourhood. By restricting this procedure to vertex orderings that are connected, we obtain connected greedy colourings. For some graphs, all connected greedy colourings

arXiv:2110.14003v1
fatcat:ycrdoiamyvarbldm55n2bblv64
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... exactly χ(G) colours; they are called good graphs. On the opposite, some graphs do not admit any connected greedy colouring using only χ(G) colours; they are called ugly graphs. We show that no perfect graph is ugly. We also give simple proofs of this fact for subclasses of perfect graphs (block graphs, comparability graphs), and show that no K_4-minor free graph is ugly.##
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Neighborhood inclusions for minimal dominating sets enumeration: linear and polynomial delay algorithms in P_7-free and P_8-free chordal graphs
[article]

2019
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arXiv
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pre-print

In [M. M. Kanté, V. Limouzy, A. Mary, and L. Nourine. On the enumeration of minimal dominating sets and related notions. SIAM Journal on Discrete Mathematics, 28(4):1916-1929, 2014] the authors give an O(n+m) delay algorithm based on neighborhood inclusions for the enumeration of minimal dominating sets in split and P_6-free chordal graphs. In this paper, we investigate generalizations of this technique to P_k-free chordal graphs for larger integers k. In particular, we give O(n+m) and O(n^3·

arXiv:1805.02412v2
fatcat:yvlfpi42snbahgvcuoukzgdmj4
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... delays algorithms in the classes of P_7-free and P_8-free chordal graphs. As for P_k-free chordal graphs for k≥ 9, we give evidence that such a technique is inefficient as a key step of the algorithm, namely the irredundant extension problem, becomes NP-complete.##
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Enumerating minimal dominating sets in K_t-free graphs and variants
[article]

2020
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arXiv
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pre-print

It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this paper we investigate this problem in graph classes defined by forbidding an induced subgraph. In particular, we provide output-polynomial time algorithms for K_t-free graphs and variants. This answers a question of Kanté et al. about enumeration in bipartite graphs.

arXiv:1810.00789v3
fatcat:taj6kwocujfxbp5gtrrsquzz3y
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