Unit Mixed Interval Graphs

Completion of the mixed unit interval graphs hierarchy

We describe the missing class of the hierarchy of mixed unit interval graphs, generated by the intersection graphs of closed, open and one type of half-open intervals of the real line. This class lies strictly between unit interval graphs and mixed unit interval graphs. We give a complete characterization of this new class, as well as quadratic-time algorithms that recognize graphs from this cl...

M1 internship report: problems on interval graphs L(2,1)-labelling Almost-mixed unit interval graphs

Integral Mixed Unit Interval Graphs

We characterize graphs that have intersection representations using unit intervals with open or closed ends such that all ends of the intervals are integral in terms of infinitely many minimal forbidden induced subgraphs. Furthermore, we provide a quadratic-time algorithm that decides if a given interval graph admits such an intersection representation.

3D-interval-filament graphs

Gavril [GA4] defined two new families of intersection graphs: the interval-filament graphs and the subtree-filament graphs. The complements of intervalfilament graphs are the cointerval mixed graphs and the complements of subtree-filament graphs are the cochordal mixed graphs. The family of interval-filament graphs contains the families of cocomparability, polygon-circle, circle and chordal gra...

A short proof that 'proper = unit'

A short proof is given that the graphs with proper interval representations are the same as the graphs with unit interval representations. An graph is an interval graph if its vertices can be assigned intervals on the real line so that vertices are adjacent if and only if the corresponding intervals intersect; such an assignment is an interval representation. When the intervals have the same le...