Constant threshold intersection graphs of orthodox paths in trees

Abstract A graph G belongs to the class ORTH [ h , s , t ] for integers h , s , and t if there is a pair ( T , S ) , where T is a tree of maximum degree at most h , and S is a collection ( S u ) u ∈ V ( G ) of subtrees S u of maximum degree at most s of T , one for each vertex u of G , such that, for every vertex u of G , all leaves of S u are also leaves of T , and, for every two distinct vertices u and v of G , the following three properties are equivalent: (i) u and v are adjacent. (ii) S u and S v have at least t nodes in common. (iii) S u and S v share a leaf of T . The class ORTH [ h , s , t ] was introduced by Jamison and Mulder. Here we focus on the case s = 2 , which is closely related to the well-known VPT and EPT graphs. We collect general properties of the graphs in ORTH [ h , 2 , t ] , and provide a characterization in terms of tree layouts. Answering a question posed by Golumbic, Lipshteyn, and Stern, we show that ORTH [ h + 1 , 2 , t ] ∖ ORTH [ h , 2 , t ] is non-empty for every h ≥ 3 and t ≥ 3 . We derive decomposition properties, which lead to efficient recognition algorithms for the graphs in ORTH [ h , 2 , 2 ] for every h ≥ 3 . Finally, we give a complete description of the graphs in ORTH [ h , 2 , 2 ] , and show that the graphs in ORTH [ 3 , 2 , 3 ] are line graphs of planar graphs.