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Many NP-hard graph problems become easy for some classes of graphs, such as coloring is easy for bipartite graphs, but NP-hard in general. So we can ask question like when does a hard problem become easy? What is the minimum substructure for which the problem remains hard? We use the notion of boundary classes to study such questions. In this paper, we introduce a method for transforming the boundary class of one NP-hard graph problem into a boundary class for another problem. If $\Pi$ and $\Gamma$ are two NP-hard graph problems where $\Pi$ is reducible to $\Gamma$, we transform a boundary class of $\Pi$ into a boundary class of $\Gamma$. More formally if $\Pi$ is reducible to $\Gamma$, where the reduction is bijective and it maps hereditary classes of graphs to hereditary classes of graphs, then $X$ is a boundary class of $\Pi$ if and only if the image of $X$ under the reduction is a boundary class of $\Gamma$. This gives us a relationship between boundary classes and reducibility among several NP-hard problems. To show the strength of our main result, we apply our theorem to obtain some previously unknown boundary classes for a few graph problems namely; vertex-cover, clique, traveling-salesperson, bounded-degree-spanning-tree, subgraph-isomorphism and clique-cover., Comment: 9 pages, 6 figures

The larger battery capacities and the longer waiting and charging time of electric vehicles (EVs) results in low utilization of charging stations (CSs). This paper, proposes fuzzy logic weight based coordination (FLWC) scheme to enhance the utilization of CSs. Each EV has an associated uncertain information including stay time and the current state-of-charge (SoC). The fuzzy logic controller (FLC) analyze these inputs and determines a weight value. The proposed FLWC scheme then allocates CSs to the EVs according to the weight values. The proposed scheme is simulated for 100 EVs and 5 CSs using Matlab. The simulation result shows about 30% improvement in the average utilization of CSs as compared to first-come-first-serve (FCFS) based scheme.

In this paper, we introduce a class of graphs which we call average hereditary graphs. Most graphs that occur in the usual graph theory applications belong to this class of graphs. Many popular types of graphs fall under this class, such as regular graphs, trees and other popular classes of graphs. We prove a new upper bound for the chromatic number of a graph in terms of its maximum average degree and show that this bound is an improvement on previous bounds. From this, we show a relationship between the average degree and the chromatic number of an average hereditary graph. This class of graphs is explored further by proving some interesting properties regarding the class of average hereditary graphs. An equivalent condition is provided for a graph to be average hereditary, through which we show that we can decide if a given graph is average hereditary in polynomial time. We then provide a construction for average hereditary graphs, using which an average hereditary graph can be recursively constructed. We also show that this class of graphs is closed under a binary operation, from this another construction is obtained for average hereditary graphs, and we see some interesting algebraic properties this class of graphs has. We then explore the effect on the complexity of graph 3-coloring problem when the input is restricted to average hereditary graphs., Comment: 12 pages, 2 figures

In this paper, we consider decision trees that use both queries based on one attribute each and queries based on hypotheses about values of all attributes. Such decision trees are similar to ones studied in exact learning, where not only membership but also equivalence queries are allowed. We investigate the problem of recognition of monotone Boolean functions with $n$ variables, $n=2, \ldots, 4$, and the problem of sorting $n$ pairwise different elements from linearly ordered set, $n=3, \ldots, 6$. For each of these problems, we compare the complexity of different types of optimal (relative to the depth or the number of realizable nodes) decision trees with hypotheses. We also study the complexity of decision trees constructed by entropy-based greedy algorithm and analyze the length of decision rules derived from these trees.

We investigate at decision trees that incorporate both traditional queries based on one attribute and queries based on hypotheses about the values of all attributes. Such decision trees are similar to ones studied in exact learning, where membership and equivalence queries are allowed. We present greedy algorithms based on diverse uncertainty measures for construction of above decision trees and discuss results of computer experiments on various data sets from the UCI ML Repository and randomly generated Boolean functions. We also study the length and coverage of decision rules derived from the decisiontrees constructed by greedy algorithms.

In this paper, we analyze static traversable wormholes via Noether symmetry technique in modified Gauss-Bonnet $f(\mathcal{G})$ theory of gravity (where $\mathcal{G}$ represents Gauss-Bonnet term). We assume isotropic matter configuration and spherically symmetric metric. We construct three $f(\mathcal{G})$ models, i.e, linear, quadratic and exponential forms and examine the consistency of these models. The traversable nature of wormhole solutions is discussed via null energy bound of the effective stress-energy tensor while physical behavior is studied through standard energy bounds of isotropic fluid. We also discuss the stability of these wormholes inside the wormhole throat and conclude the presence of traversable and physically stable wormholes for quadratic as well as exponential $f(\mathcal{G})$ models., Comment: 28 pages, 14 figures, to appear in EPJC