A Method to Identify and Analyze Biological Programs through Automated Reasoning.

Predictive biology is elusive because rigorous, data-constrained, mechanistic models of complex biological systems are difficult to derive and validate. Current approaches tend to construct and examine static interaction network models, which are descriptively rich but often lack explanatory and predictive power, or dynamic models that can be simulated to reproduce known behavior. However, in such approaches implicit assumptions are introduced as typically only one mechanism is considered, and exhaustively investigating all scenarios is impractical using simulation. To address these limitations, we present a methodology based on automated formal reasoning, which permits the synthesis and analysis of the complete set of logical models consistent with experimental observations. We test hypotheses against all candidate models, and remove the need for simulation by characterizing and simultaneously analyzing all mechanistic explanations of observed behavior. Our methodology transforms knowledge of complex biological processes from sets of possible interactions and experimental observations to precise, predictive biological programs governing cell function

On the evaluation of the Tutte polynomial at the points (1,-1) and (2,-1)

C. Merino proved recently the following identity between evaluations of the Tutte polynomial of complete graphs: t($K_{n+2}$; 1,−1) = t($K_n$;2,−1). In this work we extend this result by giving a large class of graphs with this property, that is, graphs G such that there exist two vertices u and v with t(G;1,−1) = t(G−{u,v};2,−1). The class is described in terms of forbidden induced subgraphs and it contains in particular threshold graphs.Postprint (published version

Decomposition of Certain Complete Graphs and Complete Multipartite Graphs into Almost-bipartite Graphs and Bipartite Graphs

In his classical paper [14], Rosa introduced a hierarchical series of labelings called ρ, σ, β and α labeling as a tool to settle Ringel’s Conjecture which states that if T is any tree with m edges then the complete graph K2m+1 can be decomposed into 2m + 1 copies of T . Inspired by the result of Rosa [14] many researchers significantly contributed to the theory of graph decomposition using graph labeling. In this direction, in 2004, Blinco et al. [6] introduced γ-labeling as a stronger version of ρ-labeling. A function g defined on the vertex set of a graph G with n edges is called a γ-labeling if (i) g is a ρ-labeling of G, (ii) G is a tripartite graph with vertex tripartition (A, B, C) with C = {c} and ¯b ∈ B such that {¯b, c} is the unique edge joining an element of B to c, (iii) g(a) \u3c g(v) for every edge {a, v} ∈ E(G) where a ∈ A, (iv) g(c) - g(¯b) = n. Further, Blinco et al. [6] proved a significant result that the complete graph K2cn+1 can be cyclically decomposed into c(2cn + 1) copies of any γ-labeled graph with n edges, where c is any positive integer. Recently, in 2013, Anita Pasotti [4] introduced a generalisation of graceful labeling called d-divisible graceful labeling as a tool to obtain cyclic G-decompositions in complete multipartite graphs. Let G be a graph of size e = d . m. A d-divisible graceful labeling of the graph G is an injective function g : V (G) → {0, 1, 2, . . . , d(m + 1) - 1} such that {|g(u) - g(v)|/{u, v} ∈ E(G)} = {1, 2, . . . , d(m + 1) - 1}\{m + 1, 2(m + 1), . . . , (d - 1)(m + 1)}. A d-divisible graceful labeling of a bipartite graph G is called as a d-divisible α-labeling of G if the maximum value of one of the two bipartite sets is less than the minimum value of the other one. Further, Anita Pasotti [4] proved a significant result that the complete multipartite graph K (e/d +1)×2dc can be cyclically decomposed into copies of d-divisible α-labeled graph G, where e is the size of the graph G and c is any positive integer (K (e/d +1)×2dc contains e/d + 1 parts each of size 2dc). Motivated by the results of Blinco et al. [6] and Anita Pasotti [4], in this paper we prove the following results. i) For t ≥ 2, disjoint union of t copies of the complete bipartite graph Km,n, where m≥ 3, n ≥ 4 plus an edge admits γ-labeling. ii) For t ≥ 2, t-levels shadow graph of the path Pdn+1 admits d-divisible α-labeling for any admissible d and n ≥ 1. Further, we discuss related open problems