  A graph $G$ is a \emph{max point-tolerance (MPT)} graph if each vertex $v$ of
$G$ can be mapped to a \emph{pointed-interval} $(I_v, p_v)$ where $I_v$ is an
interval of $\mathbb{R}$ and $p_v \in I_v$ such that $uv$ is an edge of $G$ iff
$I_u \cap I_v \supseteq \{p_u, p_v\}$. 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
case 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 maximum weight
independent set problem can be solved in polynomial time, the coloring 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 interval graphs and outerplanar graphs, but are incomparable
to permutation, chordal, and planar graphs.
