Rita Zuazua

On Rigid Minimal Presentations

Abstract Let Λ be a basic finite dimensional k-algebra which contains a semisimple algebra S separable over k with Λ = S ⊕ J, where J is the Jacobson radical of Λ. The category 𝒫(Λ) of radical morphisms between projective left Λ-modules has an exact structure. For ϕ1, ϕ2 in 𝒫(Λ), denote by Ext𝒫(ϕ1, ϕ2) the corresponding extension group. A finitely generated left Λ-module M has rigid minimal projective presentation if Ext𝒫(ϕ, ϕ) = 0. We will see that if M is indecomposable and has rigid minimal projective presentation then EndΛ(M)/radEndΛ(M) ≅ EndΛ(T) for T a simple Λ-module.

Some Variations of Perfect Graphs

Abstract We consider (ψk−γk−1)-perfect graphs, i.e., graphs G for which ψk(H) = γk−1(H) for any induced subgraph H of G, where ψk and γk−1 are the k-path vertex cover number and the distance (k − 1)-domination number, respectively. We study (ψk−γk−1)-perfect paths, cycles and complete graphs for k ≥ 2. Moreover, we provide a complete characterisation of (ψ2 − γ1)- perfect graphs describing the set of its forbidden induced subgraphs and providing the explicit characterisation of the structure of graphs belonging to this family.

Total domination multisubdivision number of a graph

Abstract The domination multisubdivision number of a nonempty graph G was defined in [3] as the minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. Similarly we define the total domination multisubdivision number msdγt (G) of a graph G and we show that for any connected graph G of order at least two, msdγt (G) ≤ 3. We show that for trees the total domination multisubdi- vision number is equal to the known total domination subdivision number. We also determine the total domination multisubdivision number for some classes of graphs and characterize trees T with msdγt (T) = 1.

On the vertices of a 3-partite tournament not in triangles

Let T be a 3-partite tournament and F 3 ( T ) be the set of vertices of T not in triangles. We prove that, if the global irregularity of T , i g ( T ) , is one and | F 3 ( T ) | 3 , then F 3 ( T ) must be contained in one of the partite sets of T and | F 3 ( T ) | ? ? k + 1 4 ? + 1 , which implies | F 3 ( T ) | ? ? n + 5 12 ? + 1 , where k is the size of the largest partite set and n the number of vertices of T . Moreover, we give some upper bounds on the number, as well as results on the structure of said vertices within the digraph, depending on its global irregularity.

Destroying longest cycles in graphs and digraphs

In 1978, C. Thomassen proved that in any graph one can destroy all the longest cycles by deleting at most one third of the vertices. We show that for graphs with circumference k ? 8 it suffices to remove at most 1 / k of the vertices. The Petersen graph demonstrates that this result cannot be extended to include k = 9 but we show that in every graph with circumference nine we can destroy all 9-cycles by removing 1 / 5 of the vertices. We consider the analogous problem for digraphs and show that for digraphs with circumference k = 2 , 3 , it suffices to remove 1 / k of the vertices. However this does not hold for k ? 4 .

Relations between edge removing and edge subdivision concerning domination number of a graph

Let

Distance 2-domination in prisms of graphs

e

Convex universal fixers

be an edge of a connected simple graph

Auslander-Reiten duality for abelian categories

G

Almost split sequences for complexes of fixed size

. The graph obtained by removing (subdividing) an edge