NP-hard graph problems and boundary classes of graphs

The width and integer optimization on simplices with bounded minors of the constraint matrices

“…The famous k-satisfiability problem is polynomial for k ≤ 2, but is NP-complete for all k > 2. A theory, when an NP-complete graph problem becomes easier, is investigated for the family of hereditary classes in the papers [1,3,2,20,21,18,22,23,24].…”

The complexity of the 3-colorability problem in the absence of a pair of small forbidden induced subgraphs

“…It was later applied for the dominating set problem [3]. A study of boundary graph classes for some graph problems was extended in the paper of Alekseev et al [2], where the notion was formulated in its most general form. We will give the necessary definitions.…”

“…The first of them is S. It constitutes all forests with at most three leaves in each connected component. The second one is T , which is the set of line graphs of graphs in S. The paper [2] is a good survey about graph problems, for which either S or T is boundary.…”

Экстремальные Множества Графов При Решении Задачи Демаркации В Семействе Наследственно Замкнутых Классов Графов

“…Этот результат подтолкнул к попыткам дать полное описание таких классов для ряда графовых задач. В числе соответствующих исследований укажем диссертации [2,9,11] и работы [5,6,7,10]. Однако центральная проблема теории граничных классов графов (получение полного описания) до настоящего времени не была решена ни для одной задачи.…”