A steady-state grouping genetic algorithm for the rainbow spanning tree problem

Given a connected, undirected and edge-colored graph, the rainbow spanning tree (RSF) problem aims to find a rainbow spanning forest with the minimum number of rainbow trees, where a rainbow tree is a connected acyclic subgraph of the graph whose each edge is associated with a different color. This problem is $NP$-Hard and finds several applications in distinguishing among various types of connections. Being a grouping problem, this paper proposes a steady-state grouping genetic algorithm (SSGGA) for the RSF problem. To the best of our knowledge, this is the first work on steady-state grouping genetic algorithm for this problem. While keeping in view of grouping aspects of the problem, each individual, in the proposed SSGGA, is encoded as a group of rainbow trees, and accordingly a problem-specific crossover operator is designed. Moreover, SSGGA uses the idea of two steps in its replacement scheme. All such elements of SSGGA coordinate effectively and overall help in finding high quality solutions. Computational results obtained over a set of benchmark instances show that overall SSGGA, in terms of solution quality, is superior to all other existing approaches in the literature for this problem.

Vertex arboricity of graphs embedded in a surface of non-negative Euler characteristic

The vertex arboricity [Formula: see text] of a graph [Formula: see text] is the minimum number of colors the vertices of the graph [Formula: see text] can be colored so that every color class induces an acyclic subgraph of [Formula: see text]. There are many results on the vertex arboricity of planar graphs. In this paper, we replace planar graphs with graphs which can be embedded in a surface [Formula: see text] of Euler characteristic [Formula: see text]. We prove that for the graph [Formula: see text] which can be embedded in a surface [Formula: see text] of Euler characteristic [Formula: see text] if no [Formula: see text]-cycle intersects a [Formula: see text]-cycle, or no [Formula: see text]-cycle intersects a [Formula: see text]-cycle, then [Formula: see text] in addition to the [Formula: see text]-regular quadrilateral mesh.

Feedback Numbers of Goldberg Snark, Twisted Goldberg Snarks and Related Graphs

A subset of vertices of a graph G is called a feedback vertex set of G if its removal results in an acyclic subgraph. The minimum cardinality of a feedback vertex set is called the feedback number. In this paper, we determine the exact values of the feedback numbers of the Goldberg snarks Gn and its related graphs Gn*, Twisted Goldberg Snarks TGn and its related graphs TGn*. Let f(n) denote the feedback numbers of these graphs, we prove that f(n)=2n+1, for n≥3.

Acyclic subgraph is hard

We prove the hardness of yet another problem in graph theory, namely Acyclic Subgraph. A reduction from Independent Set shows that our claim holds.

Listing all spanning trees in Halin graphs — sequential and Parallel view

For a connected labeled graph [Formula: see text], a spanning tree [Formula: see text] is a connected and acyclic subgraph that spans all vertices of [Formula: see text]. In this paper, we consider a classical combinatorial problem which is to list all spanning trees of [Formula: see text]. A Halin graph is a graph obtained from a tree with no degree two vertices and by joining all leaves with a cycle. We present a sequential and parallel algorithm to enumerate all spanning trees in Halin graphs. Our approach enumerates without repetitions and we make use of [Formula: see text] processors for parallel algorithmics, where [Formula: see text] and [Formula: see text] are the depth, the number of leaves, respectively, of the Halin graph. We also prove that the number of spanning trees in Halin graphs is [Formula: see text].

Trees, Trees, So Many Trees

This chapter considers the problem of counting trees. Every connected graph G has a spanning tree, that is, a connected acyclic subgraph containing all the vertices of G. If G has no cycles, it is its own unique spanning tree. If G has cycles, we can locate any cycle and delete one of its edges. Repeat this process until no cycle remains. We have just constructed one of the spanning trees of G. Typically G will have many, many spanning trees. Let us use t(G) to denote the number of spanning trees in G. There are several ways to determine t(G). Some of these are direct argument, Kirchhoff's Matrix Tree Theorem, a variation of this theorem using eigenvalues, and Prüfer codes.

Superbubbles, Ultrabubbles and Cacti

A superbubble is a type of directed acyclic subgraph with single distinct source and sink vertices. In genome assembly and genetics, the possible paths through a superbubble can be considered to represent the set of possible sequences at a location in a genome. Bidirected and biedged graphs are a generalization of digraphs that are increasingly being used to more fully represent genome assembly and variation problems. Here we define snarls and ultrabubbles, generalizations of superbubbles for bidirected and biedged graphs, and give an efficient algorithm for the detection of these more general structures. Key to this algorithm is the cactus graph, which we show encodes the nested decomposition of a graph into snarls and ultrabubbles within its structure. We propose and demonstrate empirically that this decomposition on bidirected and biedged graphs solves a fundamental problem by defining genetic sites for any collection of genomic variations, including complex structural variations, without need for any single reference genome coordinate system. Furthermore, the nesting of the decomposition gives a natural way to describe and model variations contained within large variations, a case not currently dealt with by existing formats, e.g. VCF.

Acyclic Subgraphs of Planar Digraphs

An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on $n$ vertices without directed 2-cycles possesses an acyclic set of size at least $3n/5$. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if $g$ is the length of the shortest directed cycle, we show that there exists an acyclic set of size at least $(1 - 3/g)n$.