A graph-theoretical approach to DNA similarity analysis

One of the very active research areas in bioinformatics is DNA similarity analysis. There are several approaches using alignment-based or alignment-free methods to analyze similarities/dissimilarities between DNA sequences. In this work, we introduce a novel representation of DNA sequences, using n-ary Cartesian products of graphs for arbitrary positive integers n. Each of the component graphs in the representing Cartesian product of each DNA sequence contain combinatorial information of certain tuples of nucleotides appearing in the DNA sequence. We further introduce a metric space structure to the set of all Cartesian products of graphs that represent a given collection of DNA sequences in order to be able to compare different Cartesian products of graphs, which in turn signifies similarities/dissimilarities between DNA sequences. We test our proposed method on several datasets including Human Papillomavirus, Human rhinovirus, Influenza A virus, and Mammals. We compare our method to other methods in literature, which indicates that our analysis results are comparable in terms of time complexity and high accuracy, and in one dataset, our method performs the best in comparison with other methods.

TWO RESULTS ON THE PALETTE INDEX OF GRAPHS

Given a proper edge coloring $\alpha$ of a graph $G$, we define the palette $S_G(v,\alpha)$ of a vertex $v\in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check{s}(G)$ of $G$ is the minimum number of distinct palettes occurring in a proper edge coloring of $G$. A graph $G$ is called nearly bipartite if there exists $ v\in V(G)$ so that $G-v$ is a bipartite graph. In this paper, we give an upper bound on the palette index of a nearly bipartite graph $G$ by using the decomposition of $G$ into cycles. We also provide an upper bound on the palette index of Cartesian products of graphs. In particular, we show that for any graphs $G$ and $H$, $\check{s}(G\square H)\leq \check{s}(G)\check{s}(H)$.

Peg Solitaire on Cartesian Products of Graphs

In 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.

Separation of Cartesian products of graphs into several connected components by the removal of edges

Let G = (V (G),E(G)) be a graph. A set S ? E(G) is an edge k-cut in G if the graph G-S = (V (G), E(G) \ S) has at least k connected components. The generalized k-edge connectivity of a graph G, denoted as ?k(G), is the minimum cardinality of an edge k-cut in G. In this article we determine generalized 3-edge connectivity of Cartesian product of connected graphs G and H and describe the structure of any minimum edge 3-cut in G2H. The generalized 3-edge connectivity ?3(G2H) is given in terms of ?3(G) and ?3(H) and in terms of other invariants of factors G and H.

On the Gonality of Cartesian Products of Graphs

In this paper we provide the first systematic treatment of Cartesian products of graphs and their divisorial gonality, which is a tropical version of the gonality of an algebraic curve defined in terms of chip-firing. We prove an upper bound on the gonality of the Cartesian product of any two graphs, and determine instances where this bound holds with equality, including for the $m\times n$ rook's graph with $\min\{m,n\}\leq 5$. We use our upper bound to prove that Baker's gonality conjecture holds for the Cartesian product of any two graphs with two or more vertices each, and we determine precisely which nontrivial product graphs have gonality equal to Baker's conjectural upper bound. We also extend some of our results to metric graphs.