A tight relation between series--parallel graphs and bipartite distance hereditary graphs

Bandelt and Mulder’s structural characterization of bipartite distance hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by re17 peatedly adding either pendant vertices or twins (i.e., vertices with the same neighborhood as an existing one). Dirac and Duffin’s structural characterization of 2–connected series–parallel graphs asserts that such graphs can be built inductively starting from a single edge by adding either edges in series or in parallel. In this paper we give an elementary proof that the two constructions are the same construction when bipartite graphs are viewed as the fundamental graphs of a graphic matroid. We then apply the result to re-prove known results concerning bipartite distance hereditary graphs and series–parallel graphs and to provide a new class of polynomially-solvable instances for the integer multi-commodity flow of maximum valu

Symetries and asymetries of the immune system response:A categorification approach

A new modeling approach and conceptual framework to the immune system response and its dual role with respect to cancer is proposed based on Applied Category Theory. States of cells and pathogenes are structured as mathematical structures (categories), the interactions, at a given phase, between cells of the immune system and pathogenes, correspond to a pair of adjunctions (adjoint functors), the interaction process consisting of the sequential composition of an identification phase, a preparation phase and an activation phase is modeled by the composition of maps of adjunctions: the approach is illustrated by considering the Cancer-Immunity Cycle. A third dimension is needed to model Cancer Immunoediting. The categorical foundations of our approach is based on Marco Grandis and Rober Paré theory of Intercategories

Polynomial functors and opetopes

We give an elementary and direct combinatorial definition of opetopes in terms of trees, well-suited for graphical manipulation and explicit computation. To relate our definition to the classical definition, we recast the Baez–Dolan slice construction for operads in terms of polynomial monads: our opetopes appear naturally as types for polynomial monads obtained by iterating the Baez–Dolan construction, starting with the trivial monad. We show that our notion of opetope agrees with Leinster's. Next we observe a suspension operation for opetopes, and define a notion of stable opetopes. Stable opetopes form a least fixpoint for the Baez–Dolan construction. A final section is devoted to example computations, and indicates also how the calculus of opetopes is well-suited for machine implementation.48 page(s