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Urban sprawl is brought on by migration brought on by the new and exciting prospects in the cities. Cities expand rapidly, which leads to increased population growth and exceeding vehicle expansion. To better understand the features and advantages of Metro Transit surveys on car ownership transit use frequency and trip purpose are crucial. These surveys are nothing more than case studies of specific transit operators from all throughout the country. Case studies explore the various city conditions the role of MRTS problems which are created by them problems solved by them and to envisage the future problems over a period of time. They also provide the clarity of the various issues and challenges faced by the city at different time of the day. Bengaluru is an outstanding example how a pensioners paradise has resulted now as a Meta city. It has undergone a huge metamorphosis trying to solve the various demands put forth by the economic activities. Transit oriented development TOD seems the logical way to address the traffic and transportation problems which results in a mixed-use development high rise high density. After Delhi and Hyderabad Bengaluru has entered the queue to embrace Namma Metro and implement Phases 2A and 2B after Phases 1 and 2. An inventory into selected case studies will relate and differentiate the conditions and systems in Bengaluru thus enriching our capability to tackle the Transit Oriented Development TOD issues at bay. Key words Public Transportation Metro rail Case study Namma Metro Bengaluru TOD, Comment: 19 pages, 4 Tables, 3 graphs and 13 chi graphs

We present an approximation algorithm for the maximum independent set (MIS) problem over the class of equilateral $B_1$-VPG graphs. These are intersection graphs of $L$-shaped planar objects % (and their rotations by multiples of $90^o$) with both arms of each object being equal. We obtain a $36(\log 2d)$-approximate algorithm running in $O(n(\log n)^2)$ time for this problem, where $d$ is the ratio $d_{max}/d_{min}$ and $d_{max}$ and $d_{min}$ denote respectively the maximum and minimum length of any arm in the input equilateral $L$-representation of the graph. In particular, we obtain $O(1)$-factor approximation of MIS for $B_1$-VPG -graphs for which the ratio $d$ is bounded by a constant. % formed by unit length $L$-shapes. In fact, algorithm can be generalized to an $O(n(\log n)^2)$ time and a $36(\log 2d_x)(\log 2d_y)$-approximate MIS algorithm over arbitrary $B_1$-VPG graphs. Here, $d_x$ and $d_y$ denote respectively the analogues of $d$ when restricted to only horizontal and vertical arms of members of the input. This is an improvement over the previously best $n^\epsilon$-approximate algorithm \cite{FoxP} (for some fixed $\epsilon>0$), unless the ratio $d$ is exponentially large in $n$. In particular, $O(1)$-approximation of MIS is achieved for graphs with $\max\{d_x,d_y\}=O(1)$.

The acyclic chromatic index, denoted by $a'(G)$, of a graph $G$ is the minimum number of colours used in any proper edge colouring of $G$ such that the union of any two colour classes does not contain a cycle, that is, forms a forest. We show that $a'(G\Box H)\le a'(G) + a'(H)$ for any two graphs $G$ and $H$ such that $max\{a'(G), a'(H)\} > 1$. Here, $G \Box H$ denotes the cartesian product of $G$ and $H$. This extends a recent result of [15] where tight and constructive bounds on $a'(G)$ were obtained for a class of grid-like graphs which can be expressed as the cartesian product of a number of paths and cycles.

We analyze a network formation game in a strategic setting where payoffs of individuals depend only on their immediate neighbourhood. We call these payoffs as localized payoffs. In this game, the payoff of each individual captures (1) the gain from immediate neighbors, (2) the bridging benefits, and (3) the cost to form links. This implies that the payoff of each individual can be computed using only its single-hop neighbourhood information. Based on this simple model of network formation, our study explores the structure of networks that form, satisfying one or both of the properties, namely, pairwise stability and efficiency. We analytically prove the pairwise stability of several interesting network structures, notably, the complete bi-partite network, complete equi-k-partite network, complete network and cycle network, under various configurations of the model. We validate and extend these results through extensive simulations. We characterize topologies of efficient networks by drawing upon classical results from extremal graph theory and discover that the Turan graph (or the complete equi-bi-partite network) is the unique efficient network under many configurations of parameters. We examine the tradeoffs between topologies of pairwise stable networks and efficient networks using the notion of price of stability, which is the ratio of the sum of payoffs of the players in an optimal pairwise stable network to that of an efficient network. Interestingly, we find that price of stability is equal to 1 for almost all configurations of parameters in the proposed model; and for the rest of the configurations of the parameters, we obtain a lower bound of 0.5 on the price of stability. This leads to another key insight of this paper: under mild conditions, efficient networks will form when strategic individuals choose to add or delete links based on only localized payoffs.

We obtain new lower bounds for the independence number of $K_r$-free graphs and linear $k$-uniform hypergraphs in terms of the degree sequence. This answers some old questions raised by Caro and Tuza \cite{CT91}. Our proof technique is an extension of a method of Caro and Wei \cite{CA79, WE79}, and we also give a new short proof of the main result of \cite{CT91} using this approach. As byproducts, we also obtain some non-trivial identities involving binomial coefficients.