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We describe the missing class of the hierarchy of mixed unit interval graphs. This class is generated

A T -decomposition of a graph G is a set of edge-disjoint copies of T in G that cover the edge set of G. Graham and Häggkvist (1989) conjectured that any 2ℓ-regular graph G admits a T -decomposition if T is a tree with ℓ edges. Kouider and Lonc (1999) conjectured that, in the special case where T is the path with ℓ edges, G admits a T -decomposition D where every vertex of G is the end-vertex of exactly two paths of D, and proved that this statement holds when G has girth at least (ℓ + 3)/2. In this paper we verify Kouider and Lonc's Conjecture for paths of length 4.

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In this text, we prove the existence of an asymptotic growth rate of the number of dominating sets (and variants) on finite rectangular grids, when the dimensions of the grid grow to infinity. Moreover, we provide, for each of the variants, an algorithm which computes the growth rate. We also give bounds on these rates provided by a computer program.4. In Section 5, we provide some bounds for the growth rates obtained by a computer program. * ).With the same method as for the lower bound, we obtain the upper bound.Remark 2. With further numerical manipulations, we notice that the lower bound and the upper bound seem to get closer to each other rather slowly. To speed up the convergence, we had the idea of using the sequences of ratios h(X D,m+1 )/h(X D,m ) and h(X D,m+1 * )/h(X D,m * ). This seems to offer a much better convergence speed. Indeed, for both sequences, from m = 11 on, the ratio seem to be stabilised around 1.954751195.