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To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let $mathcal{G}(g,c,n)$ be the set of graphs $G$ with girth $g(G)=g$, circumference $c(G)=c$, and $n$ vertices; and let $mathcal{H}(g,c,m)$ be the set of graphs with girth $g$, circumference $c$, and $m$ edges. In this work, we study the four following extremal problems on graphs: $A(g,c,n)=min{delta(G),|; G in mathcal{G}(g,c,n) }$, $B(g,c,n)=max{delta(G),|; G in mathcal{G}(g,c,n) }$, $alpha(g,c,m)=min{delta(G),|; in mathcal{H}(g,c,m) }$ and $beta(g,c,m)=max{delta(G),|; G in mathcal{H}(g,c,m) }$. In particular, we obtain bounds for $A(g,c,n)$ and $alpha(g,c,m)$, and we compute the precise value of $B(g,c,n)$ and $beta(g,c,m)$ for all values of $g$, $c$, $n$ and $m$.
Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. It measures the tree-likeness of a graph from a metric viewpoint. In particular, we are interested in circular-arc graphs,
The general position number of a graph $G$ is the size of the largest set of vertices $S$ such that no geodesic of $G$ contains more than two elements of $S$. The monophonic position number of a graph is defined similarly, but with `induced path in p
If $X$ is a geodesic metric space and $x_{1},x_{2},x_{3} in X$, a geodesic triangle $T={x_{1},x_{2},x_{3}}$ is the union of the three geodesics $[x_{1}x_{2}]$, $[x_{2}x_{3}]$ and $[x_{3}x_{1}]$ in $X$. The space $X$ is $delta$-hyperbolic in the Gromo
Assume $ k $ is a positive integer, $ lambda={k_1,k_2,...,k_q} $ is a partition of $ k $ and $ G $ is a graph. A $lambda$-assignment of $ G $ is a $ k $-assignment $ L $ of $ G $ such that the colour set $ bigcup_{vin V(G)} L(v) $ can be partitioned
A class of simple graphs such as ${cal G}$ is said to be {it odd-girth-closed} if for any positive integer $g$ there exists a graph $G in {cal G}$ such that the odd-girth of $G$ is greater than or equal to $g$. An odd-girth-closed class of graphs ${c