The (a,b,s,t)-diameter of graphs: a particular case of conditional diameter

The conditional diameter of a connected graph $\Gamma=(V,E)$ is defined as follows: given a property ${\cal P}$ of a pair $(\Gamma_1, \Gamma_2)$ of subgraphs of $\Gamma$, the so-called \emph{conditional diameter} or ${\cal P}$-{\em diameter} measures the maximum distance among subgraphs satisfying ${\cal P}$. That is, $$D_{{\cal P}}(\Gamma):=\max_{\Gamma_1, \Gamma_2\subset \Gamma} \{\partial(\Gamma_1, \Gamma_2): \Gamma_1, \Gamma_2 \quad {\rm satisfy }\quad {\cal P}\}.$$ In this paper we consider the conditional diameter in which ${\cal P}$ requires that $\delta(u)\ge \alpha$ for all $u\in V(\Gamma_1)$, $\delta(v)\ge \beta$ for all $v\in V(\Gamma_2)$, $| V(\Gamma_1)| \ge s$ and $| V(\Gamma_2)| \ge t$ for some integers $1\le s,t\le |V|$ and $\delta \le \alpha, \beta \le \Delta$, where $\delta(x)$ denotes the degree of a vertex $x$ of $\Gamma$, $\delta$ denotes the minimum degree and $\Delta$ the maximum degree of $\Gamma$. The conditional diameter obtained is called $(\alpha ,\beta, s,t)$-\emph{diameter}. We obtain upper bounds on the $(\alpha ,\beta, s,t)$-diameter by using the $k$-alternating polynomials on the mesh of eigenvalues of an associated weighted graph. The method provides also bounds for other parameters such as vertex separators

On the acyclic disconnection and the girth

The acyclic disconnection, (omega) over right arrow (D), of a digraph D is the maximum number of connected components of the underlying graph of D - A(D*), where D* is an acyclic subdigraph of D. We prove that (omega) over right arrow (D) >= g - 1 for every strongly connected digraph with girth g >= 4, and we show that (omega) over right arrow (D) = g - 1 if and only if D congruent to C-g for g >= 5. We also characterize the digraphs that satisfy (omega) over right arrow (D) = g - 1, for g = 4 in certain classes of digraphs. Finally, we define a family of bipartite tournaments based on projective planes and we prove that their acyclic disconnection is equal to 3. Then, these bipartite tournaments are counterexamples of the conjecture (omega) over right arrow (T) = 3 if and only if T congruent to (C) over right arrow (4) posed for bipartite tournaments by Figueroa et al. (2012). (C) 2015 Elsevier B.V. All rights reserved.Peer ReviewedPostprint (author's final draft