Problems on Matchings and Independent Sets of a Graph

Let $G$ be a finite simple graph. For $X \subset V(G)$, the difference of $X$, $d(X) := |X| - |N (X)|$ where $N(X)$ is the neighborhood of $X$ and $\max \, \{d(X):X\subset V(G)\}$ is called the critical difference of $G$. $X$ is called a critical set if $d(X)$ equals the critical difference and ker$(G)$ is the intersection of all critical sets. It is known that ker$(G)$ is an independent (vertex) set of $G$. diadem$(G)$ is the union of all critical independent sets. An independent set $S$ is an inclusion minimal set with $d(S) > 0$ if no proper subset of $S$ has positive difference. A graph $G$ is called K\"onig-Egerv\'ary if the sum of its independence number ($\alpha (G)$) and matching number ($\mu (G)$) equals $|V(G)|$. It is known that bipartite graphs are K\"onig-Egerv\'ary. In this paper, we study independent sets with positive difference for which every proper subset has a smaller difference and prove a result conjectured by Levit and Mandrescu in 2013. The conjecture states that for any graph, the number of inclusion minimal sets $S$ with $d(S) > 0$ is at least the critical difference of the graph. We also give a short proof of the inequality $|$ker$(G)| + |$diadem$(G)| \le 2\alpha (G)$ (proved by Short in 2016). A characterization of unicyclic non-K\"onig-Egerv\'ary graphs is also presented and a conjecture which states that for such a graph $G$, the critical difference equals $\alpha (G) - \mu (G)$, is proved. We also make an observation about ker$G)$ using Edmonds-Gallai Structure Theorem as a concluding remark.Comment: 18 pages, 2 figure

Cones of closed alternating walks and trails

Consider a graph whose edges have been colored red and blue. Assign a nonnegative real weight to every edge so that at every vertex, the sum of the weights of the incident red edges equals the sum of the weights of the incident blue edges. The set of all such assignments forms a convex polyhedral cone in the edge space, called the \emph{alternating cone}. The integral (respectively, $\{0,1\}$) vectors in the alternating cone are sums of characteristic vectors of closed alternating walks (respectively, trails). We study the basic properties of the alternating cone, determine its dimension and extreme rays, and relate its dimension to the majorization order on degree sequences. We consider whether the alternating cone has integral vectors in a given box, and use residual graph techniques to reduce this problem to searching for a closed alternating trail through a given edge. The latter problem, called alternating reachability, is solved in a companion paper along with related results.Comment: Minor rephrasing, new pictures, 14 page

Thermodynamics of Morpholine Complexes of Co(II), Cu(II) & Zn(II) Fluoborates-A Study in Solid Phase

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The polytope of dual degree partitions

AbstractWe determine the extreme points and facets of the convex hull of all dual degree partitions of simple graphs on n vertices. (This problem was raised in the Laplace Energy group of the Workshop Spectra of Families of Matrices described by Graphs, Digraphs, and Sign Patterns held at the American Institute of Mathematics Research Conference Center on October 23–27, 2006 [R. Brualdi, Leslie Hogben, Brian Shader, AIM Workshop – Spectra of Families of Matrices Described by Graphs, Digraphs, and Sign Patterns, Final Report: Mathematical Results, November 17, 2006].

Online Algorithms with Discrete Visibility - Exploring Unknown Polygonal Environments

The context of this work is the exploration of unknown polygonal environments with obstacles. Both the outer boundary and the boundaries of obstacles are piecewise linear. The boundaries can be nonconvex. The exploration problem can be motivated by the following application. Imagine that a robot has to explore the interior of a collapsed building, which has crumbled due to an earthquake, to search for human survivors. It is clearly impossible to have a knowledge of the building's interior geometry prior to the exploration. Thus, the robot must be able to see, with its onboard vision sensors, all points in the building's interior while following its exploration path. In this way, no potential survivors will be missed by the exploring robot. The exploratory path must clearly reflect the topology of the free space, and, therefore, such exploratory paths can be used to guide future robot excursions (such as would arise in our example from a rescue operation)

A GEANT-based study of atmospheric neutrino oscillation parameters at INO

We have studied the dependence of the allowed space of the atmospheric neutrino oscillation parameters on the time of exposure for a magnetized Iron CALorimeter (ICAL) detector at the India-based Neutrino Observatory (INO). We have performed a Monte Carlo simulation for a 50 kTon ICAL detector generating events by the neutrino generator NUANCE and simulating the detector response by GEANT. A chi-square analysis for the ratio of the up-going and down-going neutrinos as a function of $L/E$ is performed and the allowed regions at 90% and 99% CL are displayed. These results are found to be better than the current experimental results of MINOS and Super-K. The possibilities of further improvement have also been discussed.Comment: 8 pages, 13 figures, a new figure added, version accepted in IJMP