Menger’s theorem for m-polar fuzzy graphs and application of m-polar fuzzy edges to road network

The main objective of this research article is to classify different types of m-polar fuzzy edges in an m-polar fuzzy graph by using the strength of connectedness between pairs of vertices. The identification of types of m-polar fuzzy edges, including α-strong m-polar fuzzy edges, β-strong m-polar fuzzy edges and δ-weak m-polar fuzzy edges proved to be very useful to completely determine the basic structure of m-polar fuzzy graph. We analyze types of m-polar fuzzy edges in strongest m-polar fuzzy path and m-polar fuzzy cycle. Further, we define various terms, including m-polar fuzzy cut-vertex, m-polar fuzzy bridge, strength reducing set of vertices and strength reducing set of edges. We highlight the difference between edge disjoint m-polar fuzzy path and internally disjoint m-polar fuzzy path from one vertex to another vertex in an m-polar fuzzy graph. We define strong size of an m-polar fuzzy graph. We then present the most celebrated result due to Karl Menger for m-polar fuzzy graphs and illustrate the vertex version of Menger’s theorem to find out the strongest m-polar fuzzy paths between affected and non-affected cities of a country due to an earthquake. Moreover, we discuss an application of types of m-polar fuzzy edges to determine traffic-accidental zones in a road network. Finally, a comparative analysis of our research work with existing techniques is presented to prove its applicability and effectiveness.

Structure and Substructure Connectivity of Hypercube-Like Networks

The connectivity of a graph [Formula: see text], [Formula: see text], is the minimum number of vertices whose removal disconnects [Formula: see text], and the value of [Formula: see text] can be determined using Menger’s theorem. It has long been one of the most important factors that characterize both graph reliability and fault tolerability. Two extensions to the classic notion of connectivity were introduced recently: structure connectivity and substructure connectivity. Let [Formula: see text] be isomorphic to any connected subgraph of [Formula: see text]. The [Formula: see text]-structure connectivity of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum set [Formula: see text] of connected subgraphs in [Formula: see text] such that every element of [Formula: see text] is isomorphic to [Formula: see text], and the removal of [Formula: see text] disconnects [Formula: see text]. The [Formula: see text]-substructure connectivity of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum set [Formula: see text] of connected subgraphs in [Formula: see text] whose removal disconnects [Formula: see text] and every element of [Formula: see text] is isomorphic to a connected subgraph of [Formula: see text]. The family of hypercube-like networks includes many well-defined network architectures, such as hypercubes, crossed cubes, twisted cubes, and so on. In this paper, both the structure and substructure connectivity of hypercube-like networks are studied with respect to the [Formula: see text]-star [Formula: see text] structure, [Formula: see text], and the [Formula: see text]-cycle [Formula: see text] structure. Moreover, we consider the relationships between these parameters and other concepts.

Countable Menger's Theorem with Finitary Matroid Constraints on the Ingoing Edges

We present a strengthening of the countable Menger's theorem of R. Aharoni. Let $ D=(V,A) $ be a countable digraph with $ s\neq t\in V $ and let $\mathcal{M}=\bigoplus_{v\in V}\mathcal{M}_v $ be a matroid on $ A $ where $ \mathcal{M}_v $ is a finitary matroid on the ingoing edges of $ v $. We show that there is a system of edge-disjoint $ s \rightarrow t $ paths $ \mathcal{P} $ such that the united edge set of these paths is $ \mathcal{M} $-independent, and there is a $ C \subseteq A $ consisting of one edge from each element of $\mathcal{P} $ for which $ \mathsf{span}_{\mathcal{M}}(C) $ covers all the $ s\rightarrow t $ paths in $ D $.

Directed Fuzzy Networks as a Model to Illicit Flows and Max Flow Min Cut Theorem

Directed fuzzy networks are introduced in this paper. They are normalized node capacitated networks and provide a good platform to model different types of complicated flows in nature. A directed fuzzy network version of Menger’s theorem and the celebrated Max flow Min cut theorem are also provided. Since the maximum flow through minimum number of directed internally disjoint paths is important in quality of service (QoS) problems in networking, the results in this paper can be applied to a wide variety of problems.

Partial impredicativity in reverse mathematics

In reverse mathematics, it is possible to have a curious situation where we know that an implication does not reverse, but appear to have no information on how to weaken the assumption while preserving the conclusion (other than reducing all the way to the tautology of assuming the conclusion). A main cause of this phenomenon is the proof of a sentence from the theory . Using methods based on the functional interpretation, we introduce a family of weakenings of and use them to give new upper bounds for the Nash-Williams Theorem of wqo theory and Menger's Theorem for countable graphs.

Plünnecke's Inequality

Plünnecke's inequality is a standard tool for obtaining estimates on the cardinality of sumsets and has many applications in additive combinatorics. We present a new proof. The main novelty is that the proof is completed with no reference to Menger's theorem or Cartesian products of graphs. We also investigate the sharpness of the inequality and show that it can be sharp for arbitrarily long, but not for infinite commutative graphs. A key step in our investigation is the construction of arbitrarily long regular commutative graphs. Lastly we prove a necessary condition for the inequality to be attained.