Complexity of Coloring Graphs without Paths and Cycles

Let $P_t$ and $C_\ell$ denote a path on $t$ vertices and a cycle on $\ell$ vertices, respectively. In this paper we study the $k$-coloring problem for $(P_t,C_\ell)$-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada, have proved that 3-colorability of $P_5$-free graphs has a finite forbidden induced subgraphs characterization, while Hoang, Moore, Recoskie, Sawada, and Vatshelle have shown that $k$-colorability of $P_5$-free graphs for $k \geq 4$ does not. These authors have also shown, aided by a computer search, that 4-colorability of $(P_5,C_5)$-free graphs does have a finite forbidden induced subgraph characterization. We prove that for any $k$, the $k$-colorability of $(P_6,C_4)$-free graphs has a finite forbidden induced subgraph characterization. We provide the full lists of forbidden induced subgraphs for $k=3$ and $k=4$. As an application, we obtain certifying polynomial time algorithms for 3-coloring and 4-coloring $(P_6,C_4)$-free graphs. (Polynomial time algorithms have been previously obtained by Golovach, Paulusma, and Song, but those algorithms are not certifying); To complement these results we show that in most other cases the $k$-coloring problem for $(P_t,C_\ell)$-free graphs is NP-complete. Specifically, for $\ell=5$ we show that $k$-coloring is NP-complete for $(P_t,C_5)$-free graphs when $k \ge 4$ and $t \ge 7$; for $\ell \ge 6$ we show that $k$-coloring is NP-complete for $(P_t,C_\ell)$-free graphs when $k \ge 5$, $t \ge 6$; and additionally, for $\ell=7$, we show that $k$-coloring is also NP-complete for $(P_t,C_7)$-free graphs if $k = 4$ and $t\ge 9$. This is the first systematic study of the complexity of the $k$-coloring problem for $(P_t,C_\ell)$-free graphs. We almost completely classify the complexity for the cases when $k \geq 4, \ell \geq 4$, and identify the last three open cases

List coloring in the absence of a linear forest.

The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The Listk-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u)⊆{1,…,k}. Let Pn denote the path on n vertices, and G+H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that Listk-Coloring can be solved in polynomial time for graphs with no induced rP1+P5, hereby extending the result of Hoàng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P5. Our result is tight; we prove that for any graph H that is a supergraph of P1+P5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H

Exhaustive generation of kk-critical H\mathcal H-free graphs

We describe an algorithm for generating all $k$-critical $\mathcal H$-free graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove that there are only finitely many $4$-critical $(P_7,C_k)$-free graphs, for both $k=4$ and $k=5$. We also show that there are only finitely many $4$-critical graphs $(P_8,C_4)$-free graphs. For each case of these cases we also give the complete lists of critical graphs and vertex-critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the $3$-colorability problem in the respective classes. Moreover, we prove that for every $t$, the class of 4-critical planar $P_t$-free graphs is finite. We also determine all 27 4-critical planar $(P_7,C_6)$-free graphs. We also prove that every $P_{10}$-free graph of girth at least five is 3-colorable, and determine the smallest 4-chromatic $P_{12}$-free graph of girth five. Moreover, we show that every $P_{13}$-free graph of girth at least six and every $P_{16}$-free graph of girth at least seven is 3-colorable. This strengthens results of Golovach et al.Comment: 17 pages, improved girth results. arXiv admin note: text overlap with arXiv:1504.0697

Continuing professional development:introducing the ERS International Certificate in Respiratory Sleep Medicine

Certification of specialists in respiratory sleep medicine for the purposes of continuing professional developmen

DNA adducts in fish following an oil spill exposure

On 12 December 1999, one third of the load of the Erika tanker, amounting to about 10,000 t crude oil flowed into sea waters close to the French Atlantic Coast. This oil contained polycyclic aromatic compounds (PAC) that are known to be genotoxic. Genotoxic effects induce DNA adducts formation, which can thus be used as pollution biomarkers. Here, we assessed the genotoxic impact of the “Erika” oil spill by DNA adducts detection in the liver of immature fishes (Solea solea) from four locations of the French Brittany coasts. Two months after the spill, a high amount of DNA adducts was found in samples from all locations, amounting to 92–290 DNA adduct per 109 nucleotides. Then total DNA adduct levels decreased to reach about 50 adducts per 109 nucleotides nine months after the spill. In vitro experiments using human cell cultures and fish liver microsomes evidence the genotoxicity of the Erika fuel. They also prove the formation of reactive species able to create DNA adducts. Furthermore, in vitro and in vivo DNA adducts fingerprints are similar, thus confirming that DNA adducts are a result of the oil spill

A Satisfiability-based Approach for Embedding Generalized Tanglegrams on Level Graphs

A tanglegram is a pair of trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in computational biology to compare evolutionary histories of species. In this paper we present a formulation of two related combinatorial embedding problems concerning tanglegrams in terms of CNF-formulas. The first problem is known as planar embedding and the second as crossing minimization problem. We show that our satisfiability formulation of these problems can handle a much more general case with more than two, not necessarily binary or complete, trees defined on arbitrary sets of leaves and allowed to vary their layouts

Introducing a core curriculum for respiratory sleep practitioners

The background and purpose of the HERMES (Harmonising Education in Respiratory Medicine for European Specialists) initiative has been discussed at length in previous articles [1-3]. This article aims to provide more detailed and specific insight into the process and methodology of the Sleep HERMES Task Force in developing a core curriculum in respiratory sleep medicine

New Polynomial Cases of the Weighted Efficient Domination Problem

Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete even for very restricted graph classes. In particular, the ED problem remains NP-complete for 2P3-free graphs and thus for P7-free graphs. We show that the weighted version of the problem (abbreviated WED) is solvable in polynomial time on various subclasses of 2P3-free and P7-free graphs, including (P2+P4)-free graphs, P5-free graphs and other classes. Furthermore, we show that a minimum weight e.d. consisting only of vertices of degree at most 2 (if one exists) can be found in polynomial time. This contrasts with our NP-completeness result for the ED problem on planar bipartite graphs with maximum degree 3

Open problems on graph coloring for special graph classes.

For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c:V→{1,2,…}c:V→{1,2,…} such that c(u)≠c(v)c(u)≠c(v) for every edge uv∈Euv∈E. We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring

Increased 30-Day Mortality in Very Old ICU Patients with COVID-19 Compared to Patients with Respiratory Failure without COVID-19

Purpose: The number of patients ≥ 80 years admitted into critical care is increasing. Coronavirus disease 2019 (COVID-19) added another challenge for clinical decisions for both admission and limitation of life-sustaining treatments (LLST). We aimed to compare the characteristics and mortality of very old critically ill patients with or without COVID-19 with a focus on LLST. Methods: Patients 80 years or older with acute respiratory failure were recruited from the VIP2 and COVIP studies. Baseline patient characteristics, interventions in intensive care unit (ICU) and outcomes (30-day survival) were recorded. COVID patients were matched to non-COVID patients based on the following factors: age (± 2 years), Sequential Organ Failure Assessment (SOFA) score (± 2 points), clinical frailty scale (± 1 point), gender and region on a 1:2 ratio. Specific ICU procedures and LLST were compared between the cohorts by means of cumulative incidence curves taking into account the competing risk of discharge and death. Results: 693 COVID patients were compared to 1393 non-COVID patients. COVID patients were younger, less frail, less severely ill with lower SOFA score, but were treated more often with invasive mechanical ventilation (MV) and had a lower 30-day survival. 404 COVID patients could be matched to 666 non-COVID patients. For COVID patients, withholding and withdrawing of LST were more frequent than for non-COVID and the 30-day survival was almost half compared to non-COVID patients. Conclusion: Very old COVID patients have a different trajectory than non-COVID patients. Whether this finding is due to a decision policy with more active treatment limitation or to an inherent higher risk of death due to COVID-19 is unclear.info:eu-repo/semantics/publishedVersio