We prove that for any two minor hereditary properties 𝓟₁ and 𝓟₂, such that 𝓟₂ covers 𝓟₁, and for any graph G ∈ 𝓟₂ there is a 𝓟₁-bipartition of G. Some remarks on minimal reducible bounds are also included.
In this paper, we develop a divide-and-conquer approach, called block decomposition, to solve the minimum geodetic set problem. This provides us with a unified approach for all graphs admitting blocks for which the problem of finding a minimum geodetic set containing a given set of vertices (g-extension problem) can be efficiently solved. Our method allows us to derive linear time algorithms for the minimum geodetic set problem in (a proper superclass of) block-cacti and monopolar chordal graphs....
We consider four combinatorial interpretations for the algebra of Boolean differential operators and construct, for each interpretation, a matrix representation for the algebra of Boolean differential operators.
Boolean matrices, the incidence matrices of a graph, are known not to be the (universal) matrices of a Boolean algebra. Here, we also show that their usual composition cannot make them the matrices of any algebra. Yet, later on, we "show" that it can. This seeming paradox comes from the hidden intrusion of a widespread set-theoretical (mis) definition and notation and denies its harmlessness. A minor modification of this standard definition might fix it.
Given a simple undirected weighted or unweighted graph, we try to cluster the vertex set into communities and also to quantify the robustness of these clusters. For that task, we propose a new method, called bootstrap clustering which consists in (i) defining a new clustering algorithm for graphs, (ii) building a set of graphs similar to the initial one, (iii) applying the clustering method to each of them, making a profile (set) of partitions, (iv) computing a consensus partition for this profile,...
Given a simple undirected weighted or unweighted graph, we try to cluster the vertex set into communities and also to quantify the robustness of these clusters. For that task, we propose a new method, called bootstrap clustering which consists in (i) defining a new clustering algorithm for graphs, (ii) building a set of graphs similar to the initial one, (iii) applying the clustering method to each of them, making a profile (set) of partitions, (iv) computing a consensus partition for this profile,...