A Characterization of Mixed Unit Interval Graphs

We give a complete characterization of mixed unit interval graphs, the intersection graphs of closed, open, and half-open unit intervals of the real line. This is a proper superclass of the well known unit interval graphs. Our result solves a problem posed by Dourado, Le, Protti, Rautenbach and Szwarcfiter (Mixed unit interval graphs, Discrete Math. 312, 3357-3363 (2012)).Comment: 17 pages, referees' comments adde

Necessary and sufficient optimality conditions for scheduling unit time jobs on identical parallel machines

In this paper we characterize optimal schedules for scheduling problems with parallel machines and unit processing times by providing necessary and sufficient conditions of optimality. We show that the optimality conditions for parallel machine scheduling are equivalent to detecting negative cycles in a specially defined graph. For a range of the objective functions, we give an insight into the underlying structure of the graph and specify the simplest types of cycles involved in the optimality conditions. Using our results we demonstrate that the optimality check can be performed by faster algorithms in comparison with existing approaches based on sufficient conditions

Hardness and approximation for the geodetic set problem in some graph classes

In this paper, we study the computational complexity of finding the \emph{geodetic number} of graphs. A set of vertices $S$ of a graph $G$ is a \emph{geodetic set} if any vertex of $G$ lies in some shortest path between some pair of vertices from $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality. In this paper, we prove that solving the \textsc{MGS} problem is NP-hard on planar graphs with a maximum degree six and line graphs. We also show that unless $P=NP$, there is no polynomial time algorithm to solve the \textsc{MGS} problem with sublogarithmic approximation factor (in terms of the number of vertices) even on graphs with diameter $2$. On the positive side, we give an $O\left(\sqrt[3]{n}\log n\right)$-approximation algorithm for the \textsc{MGS} problem on general graphs of order $n$. We also give a $3$-approximation algorithm for the \textsc{MGS} problem on the family of solid grid graphs which is a subclass of planar graphs

Value of adenosine infusion for infarct size determination using real-time myocardial contrast echocardiography

BACKGROUND: Myocardial contrast echocardiography has been used for determination of infarct size (IS) in experimental models. However, with intermittent harmonic imaging, IS seems to be underestimated immediately after reperfusion due to areas with preserved, yet dysfunctional, microvasculature. The use of exogenous vasodilators showed to be useful to unmask these infarcted areas with depressed coronary flow reserve. This study was undertaken to assess the value of adenosine for IS determination in an open-chest canine model of coronary occlusion and reperfusion, using real-time myocardial contrast echocardiography (RTMCE). METHODS: Nine dogs underwent 180 minutes of coronary occlusion followed by reperfusion. PESDA (Perfluorocarbon-Exposed Sonicated Dextrose Albumin) was used as contrast agent. IS was determined by RTMCE before and during adenosine infusion at a rate of 140 mcg·Kg(-1)·min(-1). Post-mortem necrotic area was determined by triphenyl-tetrazolium chloride (TTC) staining. RESULTS: IS determined by RTMCE was 1.98 ± 1.30 cm(2 )and increased to 2.58 ± 1.53 cm(2 )during adenosine infusion (p = 0.004), with good correlation between measurements (r = 0.91; p < 0.01). The necrotic area determined by TTC was 2.29 ± 1.36 cm(2 )and showed no significant difference with IS determined by RTMCE before or during hyperemia. A slight better correlation between RTMCE and TTC measurements was observed during adenosine (r = 0.99; p < 0.001) then before it (r = 0.92; p = 0.0013). CONCLUSION: RTMCE can accurately determine IS in immediate period after acute myocardial infarction. Adenosine infusion results in a slight better detection of actual size of myocardial damage