An H-magic labeling in a H-decomposable graph G is a bijection f : V (G) ∪ E(G) → {1, 2, ..., p + q} such that for every copy H in the decomposition, ΣνεV(H) f(v) +  ΣeεE(H) f(e) is constant. f is said to be H-E-super magic if f(E(G)) = {1, 2, · · · , q}. A family of subgraphs H1,H2, · · · ,Hh of G is a mixed cycle-decomposition of G if every subgraph Hi is isomorphic to some cycle Ck, for k ≥ ...

An H-magic labeling in a H-decomposable graph G is a bijection f : V (G) ∪ E(G) → {1, 2, ..., p + q} such that for every copy H in the decomposition, ∑νεV (H) f(v) + ∑νεE (H) f(e) is constant. f is said to be H-E-super magic if f(E(G)) = {1, 2, · · · , q}. A family of subgraphs H1,H2, · · · ,Hh of G is a mixed cycle-decomposition of G if every subgraph Hi is isomorphic to some cycle Ck, for k ≥...

In this paper, we design an algorithm that, given a directed graph G and the Cartesian-product decomposition of its underlying undirected graph G̃, produces the directed Cartesian-product decomposition of G in linear time. This is the first time that the linear complexity is achieved for this problem, which has two major consequences. Firstly, it shows that the directed and undirected versions o...

We prove that the unique decomposition of connected graphs defined by Tutte is definable by formulas of Monadic Second-Order Logic. This decomposition has two levels: every connected graph is a tree of "2-connected components" called blocks ; every 2-connected graph is a tree of so-called 3-blocks. Our proof uses 2dags which are certain acyclic orientations of the considered graphs. We obtain a...

In this paper, a new general decomposition theory inspired from modular graph decomposition is presented. Our main result shows that, within this general theory, most of the nice algorithmic tools developed for modular decomposition are still efficient. This theory not only unifies the usual modular decomposition generalisations such as modular decomposition of directed graphs or decomposition ...

In this paper we consider linear systems with faults. We present a new system decomposition well suited for sensor location in the Fault Detection and Isolation problem. We deal with this problem when the system under consideration is structured, that is, the entries of the system matrices are either fixed zeros or free parameters. To such structured systems one can associate a graph. We presen...

We consider the decomposition of the complete bipartite graph K'm,n into isomorphic copies of a d-cube. We present some general necessary conditions for such a decomposition and show that these conditions are sufficient for d 3 and d = 4. We also explore the d-cube decomposition of the complete graph Kn. Necessary and sufficient conditions for the existence of such a decomposition are known for...

We show that all the tangles in a finite graph or matroid can be distinguished by a single tree-decomposition that is invariant under the automorphisms of the graph or matroid. This comes as a corollary of a similar decomposition theorem for more general combinatorial structures, which has further applications.

This paper is a preliminary attempt to study how modular and bimodular decomposition, used in graph theory, can be used on contexts and concept lattices in formal concept analysis (FCA). In a graph, a module is a set of vertices defined in term of behaviour with respect to the outside of the module: All vertices in the module act with no distinction and can be replaced by a unique vertex, which...

An H-decomposition of a graph G is a set L of edge-disjoint Hsubgraphs of G, such that each edge of G appears in some element of L. A k-orthogonal H-decomposition of a graph G is a set of k H-decompositions of G, such that any two copies of H in any two distinct H-decompositions have at most one edge in common. We prove that for every fixed graph H and every fixed integer k ≥ 1, if n is suffici...