We prove the Girth Alternative for finitely generated subgroups of PLo(I). We also prove that a finitely generated subgroup of Homeo+(I) which is sufficiently rich with hyperbolic-like elements has infinite girth.

We study the problem of determining the girth of an unweighted undirected graph. We obtain several new efficient approximation algorithms for graphs with n nodes and m edges and unknown girth g. We consider additive and multiplicative approximations. Additive Approximations. We present: • an Õ(n/m)-time algorithm which returns a cycle of length at most g + 2 if g is even and g + 3 if g is odd. ...

In this paper, we obtain lower bounds for the domination numbers of connected graphs with girth at least 7. We show that the domination number of a connected graph with girth at least 7 is either 1 or at least 1 2 (3 + √ 8(m− n) + 9), where n is the number of vertices in the graph and m is the number of edges in the graph. For graphs with minimum degree 2 and girth at least 7, the lower bound c...

For each positive integer d, we present a bijection between the set of planar maps of girth d inside a d-gon and a set of decorated plane trees. The bijection has the property that each face of degree k in the map corresponds to a vertex of degree k in the tree, so that maps of girth d can be counted according to the degree distribution of their faces. More precisely, we obtain for each integer...

This article presents unified bijective constructions for planar maps, with control on the face degrees and on the girth. Recall that the girth is the length of the smallest cycle, so that maps of girth at least d = 1, 2, 3 are respectively the general, loopless, and simple maps. For each positive integer d, we obtain a bijection for the class of plane maps (maps with one distinguished root-fac...

We study graphs in which the maximum and the minimum sizes of a maximal independent set differ by exactly one. We call these graphs almost well-covered, in analogy with the class of well-covered graphs, in which all maximal independent sets have the same size. A characterization of graphs of girth at least 8 having exactly two different sizes of maximal independent sets due to Finbow, Hartnell,...

An individual distance-independent girth increment model for pure stands of Douglas-fir (Pseudotsuga menziesii (MIRB.) FRANCO), comprising two equations, is presented. The data used to fit the model were collected from 1007 trees in 42 plots installed in regularly stocked and even-aged stands located in Wallonia (southern Belgium). Both equations predict girth increment from individual girth, d...

We prove that for graphs of order n, minimum degree δ ≥ 2 and girth g ≥ 5 the domination number γ satisfies γ ≤ ( 1 3 + 2 3g ) n. As a corollary this implies that for cubic graphs of order n and girth g ≥ 5 the domination number γ satisfies γ ≤ ( 44 135 + 82 135g ) n which improves recent results due to Kostochka and Stodolsky (An upper bound on the domination number of n-vertex connected cubic...

Consider the following problem: Does there exist a graph with large girth and large chromatic number? The girth of a graph is the length of the shortest cycle and the chromatic number is the minimum number of colors needed to color the vertices of the graph so no two adjacent vertices have the same color. These two conditions seem contradictory, intuitively to have high chromatic number we need...

We provide proofs of the following theorems by considering the entropy of random walks. Theorem 1.(Alon, Hoory and Linial) Let G be an undirected simple graph with n vertices, girth g, minimum degree at least 2 and average degree d̄. Odd girth: If g = 2r + 1, then n ≥ 1 + d̄ r−1 ∑ i=0 (d̄− 1)i. Even girth: If g = 2r, then n ≥ 2 r−1 ∑ i=0 (d̄− 1)i. Theorem 2.(Hoory) Let G = (VL, VR, E) be a bipartit...