Leafy spanning k-forests

We denote by Maximum Leaf Spanning k-Forest the problem of, given a positive integer k and a graph G with at most k components, finding a spanning forest in G with at most k components and the maximum number of leaves. A leaf in a forest is defined as a vertex of degree at most one. The case k = 1 for connected graphs is known to be NP-hard, and is well studied in the literature, with the best approximation algorithm proposed more than 20 years ago by Solis-Oba. The best known approximation algorithm for Maximum Leaf Spanning k-Forest with a slightly different leaf definition is a 3-approximation based on an approach by Lu and Ravi for the k = 1 case. We extend the algorithm of Solis-Oba to achieve a 2-approximation for Maximum Leaf Spanning k-Forest.

A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

This paper has two main parts. First, we consider the Tutte symmetric function XB, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of XB and show that this function admits a deletion-contraction relation. We also demonstrate that the vertex-weighted XB admits spanning-tree and spanning-forest expansions generalizing those of the Tutte polynomial by connecting XB to other graph functions. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific examples.

Centralized Multi-Hop Routing Based on Multi-Start Minimum Spanning Forest Algorithm in the Wireless Sensor Networks

Wireless sensor networks (WSNs) are widely applied in environmental monitoring, target tracking, military, and industrial fields. However, the battery energy of sensor nodes in WSNs is limited, which limits its development. Previous studies have shown that clustering protocols and multi-hop communication are beneficial to reduce nodes energy consumption. The multi-hop protocol based on low energy adaptive clustering hierarchy (LEACH) has been proven to significantly reduce energy dissipation. However, LEACH-based multi-hop protocols generally have the problem of unbalanced energy dissipation and data conflicts. In this paper, we propose a centralized multi-hop routing based on multi-start minimum spanning forest (LEACH-CMF) to optimize LEACH. In order to realize multi-hop communication, we introduced a multi-start minimum spanning tree algorithm to select relay nodes with the minimum relay cost and generate appropriate multi-hop paths. To avoid data collision in multi-hop communication and make nodes including the cluster heads sleep as much as possible in the non-working state, we design a bottom-up continuous time slot allocation method to improve the time division multiple access (TDMA) cycle. We performed simulation in NS2. The simulation results show that the network lifetime is approximately doubled compared to LEACH and centralized low energy adaptive clustering hierarchy (LEACH-C). The simulation results show that the proposed protocol can effectively balance the energy dissipation of nodes and prolong network lifetime.

Fast marching based superpixels

In this article, we present a fast-marching based algorithm for generating superpixel (FMS) partitions of images. The idea behind the algorithm is to draw an analogy between waves propagating in a heterogeneous medium and regions growing on an image at a rate depending on the local color and texture. The FMS algorithm is evaluated on the Berkeley Segmentation Dataset 500. It yields results in terms of boundary adherence that are slightly better than the ones obtained with similar approaches including the Simple Linear Iterative Clustering, the Eikonal-based region growing for efficient clustering and the Iterative Spanning Forest framework for superpixel segmentation algorithms. An interesting feature of the proposed algorithm is that it can take into account texture information to compute the superpixel partition. We illustrate the interest of adding texture information on a specific set of images obtained by recombining textures patches extracted from images representing stripes, originally constructed by Giraud et al. [20]. On this dataset, our approach works significantly better than color based superpixel algorithms.