A linear-time algorithm for finding a complete graph minor in a dense graph

Let g(t) be the minimum number such that every graph G with average degree d(G) \geq g(t) contains a K_{t}-minor. Such a function is known to exist, as originally shown by Mader. Kostochka and Thomason independently proved that g(t) \in \Theta(t*sqrt{log t}). This article shows that for all fixed \epsilon > 0 and fixed sufficiently large t \geq t(\epsilon), if d(G) \geq (2+\epsilon)g(t) then we can find this K_{t}-minor in linear time. This improves a previous result by Reed and Wood who gave a linear-time algorithm when d(G) \geq 2^{t-2}.Comment: 6 pages, 0 figures; Clarification added in several places, no change to arguments or result

A note on the Cops & Robber game on graphs embedded in non-orientable surfaces

The Cops and Robber game is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if they can catch the robber. The minimum number of cops needed to win on a graph is called its cop number. It is known that the cop number of a graph embedded on a surface $X$ of genus $g$ is at most $3g/2 + 3$, if $X$ is orientable (Schroeder 2004), and at most $2g+1$, otherwise (Nowakowski & Schroeder 1997). We improve the bounds for non-orientable surfaces by reduction to the orientable case using covering spaces. As corollaries, using Schroeder's results, we obtain the following: the maximum cop number of graphs embeddable in the projective plane is 3; the cop number of graphs embeddable in the Klein Bottle is at most 4, and an upper bound is $3g/2 + 3/2$ for all other $g$.Comment: 5 pages, 1 figur

An Improved Bound for First-Fit on Posets Without Two Long Incomparable Chains

It is known that the First-Fit algorithm for partitioning a poset P into chains uses relatively few chains when P does not have two incomparable chains each of size k. In particular, if P has width w then Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010) proved an upper bound of ckw^{2} on the number of chains used by First-Fit for some constant c, while Joret and Milans (Order, 28(3):455--464, 2011) gave one of ck^{2}w. In this paper we prove an upper bound of the form ckw. This is best possible up to the value of c.Comment: v3: referees' comments incorporate

Design and Development of a portable Pulse Oximetry system

Heart is the most important part in human body. Thus, it is important to follow-up and monitor its condition. Heart rate (HR) and blood oxygen saturation (SpO2) are important indicators directly related to heart-pulmonary system. Monitoring of HR and SpO2 offers us a good indication of heart functionality. Therefore, it is crucial to design and develop a homemade inexpensive device for measuring HR and SpO2. Pulse Oximeter (PO) is an opto-electronic non-invasive medical instrument capable of measuring and recording the changes of HR and SpO2 at the finger tip. In this paper we will demonstrate the overall process involved in the development of a portable (PO) system which can be used for health condition monitoring or for educational and research purposes

Boxicity of graphs on surfaces

The boxicity of a graph $G=(V,E)$ is the least integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \le i \le k$, such that $E=E_1 \cap ... \cap E_k$. Scheinerman proved in 1984 that outerplanar graphs have boxicity at most two and Thomassen proved in 1986 that planar graphs have boxicity at most three. In this note we prove that the boxicity of toroidal graphs is at most 7, and that the boxicity of graphs embeddable in a surface $\Sigma$ of genus $g$ is at most $5g+3$. This result yields improved bounds on the dimension of the adjacency poset of graphs on surfaces.Comment: 9 pages, 2 figure

Trees with Given Stability Number and Minimum Number of Stable Sets

We study the structure of trees minimizing their number of stable sets for given order $n$ and stability number $\alpha$. Our main result is that the edges of a non-trivial extremal tree can be partitioned into $n-\alpha$ stars, each of size $\lceil \frac{n-1}{n-\alpha} \rceil$ or $\lfloor \frac{n-1}{n-\alpha}\rfloor$, so that every vertex is included in at most two distinct stars, and the centers of these stars form a stable set of the tree.Comment: v2: Referees' comments incorporate

Irreducible triangulations of surfaces with boundary

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was known only for surfaces without boundary (b=0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary

Assortment optimisation under a general discrete choice model: A tight analysis of revenue-ordered assortments

The assortment problem in revenue management is the problem of deciding which subset of products to offer to consumers in order to maximise revenue. A simple and natural strategy is to select the best assortment out of all those that are constructed by fixing a threshold revenue $\pi$ and then choosing all products with revenue at least $\pi$. This is known as the revenue-ordered assortments strategy. In this paper we study the approximation guarantees provided by revenue-ordered assortments when customers are rational in the following sense: the probability of selecting a specific product from the set being offered cannot increase if the set is enlarged. This rationality assumption, known as regularity, is satisfied by almost all discrete choice models considered in the revenue management and choice theory literature, and in particular by random utility models. The bounds we obtain are tight and improve on recent results in that direction, such as for the Mixed Multinomial Logit model by Rusmevichientong et al. (2014). An appealing feature of our analysis is its simplicity, as it relies only on the regularity condition. We also draw a connection between assortment optimisation and two pricing problems called unit demand envy-free pricing and Stackelberg minimum spanning tree: These problems can be restated as assortment problems under discrete choice models satisfying the regularity condition, and moreover revenue-ordered assortments correspond then to the well-studied uniform pricing heuristic. When specialised to that setting, the general bounds we establish for revenue-ordered assortments match and unify the best known results on uniform pricing.Comment: Minor changes following referees' comment

Parameterized vertex deletion problems for hereditary graph classes with a block property

For a class of graphs P, the Bounded P-Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether there is a set S of at most k vertices such that each block of G − S has at most d vertices and is in P. We show that when P satisfies a natural hereditary property and is recognizable in polynomial time, Bounded P-Block Vertex Deletion can be solved in time 2O(k log d)nO(1), and this running time cannot be improved to 2o(k log d)nO(1), in general, unless the Exponential Time Hypothesis fails. On the other hand, if P consists of only complete graphs, or only K1,K2, and cycle graphs, then Bounded P-Block Vertex Deletion admits a cknO(1)-time algorithm for some constant c independent of d. We also show that Bounded P-Block Vertex Deletion admits a kernel with O(k2d7) vertices. © Springer-Verlag GmbH Germany 2016

Circulating adrenomedullin estimates survival and reversibility of organ failure in sepsis: the prospective observational multinational Adrenomedullin and Outcome in Sepsis and Septic Shock-1 (AdrenOSS-1) study

Background: Adrenomedullin (ADM) regulates vascular tone and endothelial permeability during sepsis. Levels of circulating biologically active ADM (bio-ADM) show an inverse relationship with blood pressure and a direct relationship with vasopressor requirement. In the present prospective observational multinational Adrenomedullin and Outcome in Sepsis and Septic Shock 1 (, AdrenOSS-1) study, we assessed relationships between circulating bio-ADM during the initial intensive care unit (ICU) stay and short-term outcome in order to eventually design a biomarker-guided randomized controlled trial. Methods: AdrenOSS-1 was a prospective observational multinational study. The primary outcome was 28-day mortality. Secondary outcomes included organ failure as defined by Sequential Organ Failure Assessment (SOFA) score, organ support with focus on vasopressor/inotropic use, and need for renal replacement therapy. AdrenOSS-1 included 583 patients admitted to the ICU with sepsis or septic shock. Results: Circulating bio-ADM levels were measured upon admission and at day 2. Median bio-ADM concentration upon admission was 80.5 pg/ml [IQR 41.5-148.1 pg/ml]. Initial SOFA score was 7 [IQR 5-10], and 28-day mortality was 22%. We found marked associations between bio-ADM upon admission and 28-day mortality (unadjusted standardized HR 2.3 [CI 1.9-2.9]; adjusted HR 1.6 [CI 1.1-2.5]) and between bio-ADM levels and SOFA score (p < 0.0001). Need of vasopressor/inotrope, renal replacement therapy, and positive fluid balance were more prevalent in patients with a bio-ADM > 70 pg/ml upon admission than in those with bio-ADM ≤ 70 pg/ml. In patients with bio-ADM > 70 pg/ml upon admission, decrease in bio-ADM below 70 pg/ml at day 2 was associated with recovery of organ function at day 7 and better 28-day outcome (9.5% mortality). By contrast, persistently elevated bio-ADM at day 2 was associated with prolonged organ dysfunction and high 28-day mortality (38.1% mortality, HR 4.9, 95% CI 2.5-9.8). Conclusions: AdrenOSS-1 shows that early levels and rapid changes in bio-ADM estimate short-term outcome in sepsis and septic shock. These data are the backbone of the design of the biomarker-guided AdrenOSS-2 trial. Trial registration: ClinicalTrials.gov, NCT02393781. Registered on March 19, 2015