19 research outputs found
On Two-Path Convexity in Multipartite Tournaments
Abstract In the context of two-path convexity, we study the rank, Helly number, Radon number, Caratheodory number, and hull number for multipartite tournaments. We show the maximum Caratheodory number of a multipartite tournament is 3. We then derive tight upper bounds for rank in both general multipartite tournaments and clone-free multipartite tournaments. We show that these same tight upper bounds hold for the Helly number, Radon number, and hull number. We classify all clone-free multipartite tournaments of maximum Helly number, Radon number, hull number, and rank. Finally we determine all convexly independent sets of clone-free multipartite tournaments of maximum rank
On the hull and interval numbers of oriented graphs
In this work, for a given oriented graph , we study its interval and hull
numbers, denoted by and , respectively, in the geodetic,
and convexities. This last one, we believe to be formally
defined and first studied in this paper, although its undirected version is
well-known in the literature. Concerning bounds, for a strongly oriented graph
, we prove that and that there is a strongly
oriented graph such that . We also determine exact
values for the hull numbers in these three convexities for tournaments, which
imply polynomial-time algorithms to compute them. These results allows us to
deduce polynomial-time algorithms to compute when the
underlying graph of is split or cobipartite. Moreover, we provide a
meta-theorem by proving that if deciding whether or
is NP-hard or W[i]-hard parameterized by , for some
, then the same holds even if the underlying graph of
is bipartite. Next, we prove that deciding whether or
is W[2]-hard parameterized by , even if the
underlying graph of is bipartite; that deciding whether or is NP-complete, even if has no directed
cycles and the underlying graph of is a chordal bipartite graph; and that
deciding whether or is W[2]-hard
parameterized by , even if the underlying graph of is split. We also
argue that the interval and hull numbers in the oriented and
convexities can be computed in polynomial time for graphs of bounded tree-width
by using Courcelle's theorem
NÚMERO ENVOLTÓRIO NA CONVEXIDADE P3: RESULTADOS E APLICAÇÕES
Este artigo apresenta uma revisão sistemática da literatura sobre os resultados e aplicações do número envoltório na convexidade P3 em grafos. A determinação deste parâmetro é equivalente ao problema de se encontrar o menor número de vértices de um grafo que permitam disseminar uma informação, influência, ou contaminação, para todos os vértices do grafo. Em particular, esta revisão descreve um panorama sobre estudos teóricos e aplicados acerca do número envoltório P3 considerando a modelagem de fenômenos sociais. Os resultados mostram que o parâmetro é pouco explorado em sociologia computacional para a modelagem de fenômenos sociais. Por outro lado, com o surgimento das redes sociais, pesquisas teóricas têm sido impulsionadas nas últimas décadas. Pesquisadores têm direcionado esforços com o objetivo de contribuir para a solução de problemas relacionados à influência social e disseminação de informação. Entretanto, ainda há espaço para estudos envolvendo o número envoltório na convexidade P3
On sufficient conditions for Hamiltonicity in dense graphs
We study structural conditions in dense graphs that guarantee the existence
of vertex-spanning substructures such as Hamilton cycles. It is easy to see
that every Hamiltonian graph is connected, has a perfect fractional matching
and, excluding the bipartite case, contains an odd cycle. Our main result in
turn states that any large enough graph that robustly satisfies these
properties must already be Hamiltonian. Moreover, the same holds for embedding
powers of cycles and graphs of sublinear bandwidth subject to natural
generalisations of connectivity, matchings and odd cycles.
This solves the embedding problem that underlies multiple lines of research
on sufficient conditions for Hamiltonicity in dense graphs. As applications, we
recover and establish Bandwidth Theorems in a variety of settings including
Ore-type degree conditions, P\'osa-type degree conditions, deficiency-type
conditions, locally dense and inseparable graphs, multipartite graphs as well
as robust expanders