(1,=l)-identifying codes in digraphs and graphs

• Autores: Berenice Martínez Barona
• Directores de la Tesis: María Camino Teófila Balbuena Martínez (dir. tes.) , Cristina Dalfó Simó (codir. tes.)
• Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2020
• Idioma: español
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• Resumen

  • BaronaThe main subject of this PhD thesis is the study of (1,=l)-identifying codes in digraphs. The results presented in this work are divided into three parts. The first one focusses on the structural properties of digraphs admitting a (1,=l)-identifying code for l=2. In the second part, we deal with the study of (1,=l)-identifying codes in line digraphs. Finally, in the third part we approach the problem from an algebraic perspective.A (1,=l)-identifying code in a digraph D is a dominating subset C of vertices of D such that all distinct subsets of vertices of cardinality at most l have distinct closed in-neighbourhoods within C. In the first part of the results, we prove that if D is a digraph admitting a (1,=l)-identifying code, then l is at most the minimum in-degree of the digraph plus one. Once this upper bound is established, we give some sufficient conditions for a digraph D, with minimum in-degree at least one, to admit a (1,=l)-identifying code for l equal to the minimum in-degree of the digraph or the minimum in-degree plus one. As a corollary, a result by Laihonen published in 2008 (that states that a k-regular graph with girth at least 7 admits a (1,=k)-identifying code) is extended to any graph of minimum degree k at least 2 and girth at least 7. Moreover, we show that every 1-in-regular digraph has a (1,=2)-identifying code if and only if the girth of the digraph is at least 5. We also characterise all the 2-in-regular digraphs admitting a (1,=l)-identifying code for l=2,3.In the second part, we prove that every line digraph of minimum in-degree 1 does not admit a (1,=l)-identifying code for l=3. Then, we give a characterisation of a line digraph of a digraph different from a directed cycle of length 4 and minimum in-degree 1 admitting a (1,=2)-identifying code. The identifying number of a digraph D, is the minimum size among all the identifying codes of D. We establish for digraphs without digons (symmetric arcs) with both vertices of in-degree 1 that the identifying number of its line digraph, is lower bounded by the number of arcs of D minus the number of vertices with out-degree at least one. Thus, we show that the identifying number of the line digraph of a digraph D attains the equality when Dhavea 1-factor with minimum in-degree 2 and without digons with both vertices of in-degree 2. We conclude by giving an algorithm to provide identifying codes in oriented digraphs with minimum in-degree at least 2 and minimum out-degree at least 1.In the third part, we give some sufficient algebraic and combinatorial conditions for a 2-in-regular digraph to admit a (1,=l)-identifying code for l=2, and for l=3 by combining the results of the first part of this thesis with some algebraic results. As far as we know, it is the first time that the spectral graph theory has been applied to the identifying codes. We present a new method to obtain an upper bound on l for digraphs by considering the positive and negative entries of eigenvectors associated with a negative eigenvalue of the adjacency matrix of the digraph. Likewise, we analyse the possible scope of using the eigenvalue zero for the same purpose. The results obtained in the directed case can also be applied to graphs.