Steve Chaplick Max Point-Tolerance Graphs A graph G is a max-point-tolerance (MPT) graph if each vertex v of G can be mapped to a pointed-interval (I_v, p_v) where Iv is an interval of R and p_v āˆˆ I_v such that uv is an edge of G iļ¬€ I_u āˆ© I_v āŠ‡ {p_u, p_v}. MPT graphs model relationships among DNA fragments in genome-wide association studies as well as basic transmission problems in telecommunications. We formally introduce this graph class, characterize it, study combinatorial optimization problems on it, and relate it to several well known graph classes. We characterize MPT graphs as a special cases of several 2D geometric intersection graphs; namely, triangle, rectangle, L-shape, and line segment intersection graphs. We further characterize MPT as having certain linear orders on their vertex set. Our last characterization is that MPT graphs are precisely obtained by intersecting special pairs of interval graphs. We also show that, on MPT graphs, the weighted independent set problem can be solved in polynomial time, the colouring problem is NP-complete, and the clique cover problem has a 2-approximation. Finally, we demonstrate several connections to known graph classes; e.g., MPT graphs strictly contain of interval graphs and outerplanar graphs, but are incomparable to permutation, chordal, and planar graphs.