Many NP-complete graph problems are polynomial-time solvable on classes of bounded clique-width. Several these a hereditary class G ${\mathscr{G}}$ if they so the atoms (graphs with no clique cut-set) . Hence, we initiate systematic study into boundedness clique-width classes. A $G$ is H $H$ -free not an induced subgraph , and it ( 1 2 ) $({H}_{1},{H}_{2})$ both ${H}_{1}$ ${H}_{2}$ -free. graph...

Let G = ( V , E ) be a simple, undirected and connected graph. A dominating set S is called secure if for each u ∈ \ there exists v such that \{ }) ∪{ } of . If further the vertex unique, then perfect (PSDS). The domination number γ ps minimum cardinality Given graph positive integer k (PSDOM) problem to check whether has PSDS size at most In this paper, we prove PSDOM NP-complete split graphs,...

We study the following variant of 15 puzzle. Given a graph and two token placements on vertices, we want to find walk minimum length (if any exists) such that sequence swappings along obtains one given from other one. This problem was introduced as Sequential Token Swapping by Yamanaka et al. [JGAA 2019], who showed is intractable in general but polynomial-time solvable for trees, complete grap...

In a graph, perfect matching cut is an edge that matching. (pmc) the problem of deciding whether given graph has cut, and known to be NP-complete. We revisit show pmc remains NP-complete when restricted bipartite graphs maximum degree 3 arbitrarily large girth. Complementing this hardness result, we give two classes in which polynomial-time solvable. The first one includes claw-free without ind...

The Induced Minor problem is that of testing whether a graph G can be modified into a graph H by a sequence of vertex deletions and edge contractions. If only edge contractions are permitted, we obtain the Contractibility problem. We prove that Induced Minor is polynomial-time solvable when G is AT-free and H is fixed, i.e., not part of the input. In addition, we show that Contractiblity is pol...

For a fixed graph H, the H-Minor Edit problem takes as input a graph G and an integer k and asks whether G can be modified into H by a total of at most k edge contractions, edge deletions and vertex deletions. Replacing edge contractions by vertex dissolutions yields the H-Topological Minor Edit problem. For each problem we show polynomial-time solvable and NP-complete cases depending on the ch...

We study the following combinatorial problem. Given a planar graph G = (V,E) and a set of simple cycles C in G, find a planar embedding E of G such that the number of cycles in C that bound a face in E is maximized. We establish a tight border of tractability for this problem in biconnected planar graphs by giving conditions under which the problem is NP-hard and showing that relaxing any of th...

A parallel knock-out scheme for a graph proceeds in rounds in each of which each surviving vertex eliminates one of its surviving neighbours. A graph is KO-reducible if there exists such a scheme that eliminates every vertex in the graph. The Parallel Knock-Out problem is to decide whether a graph G is KO-reducible. This problem is known to be NP-complete and has been studied for several graph ...

Precoloring Extension (shortly PrExt) is the following problem: Given a graph G with some precolored vertices and a color bound k, can the precoloring of G be extended to a proper coloring of all vertices of G using not more than k colors? Answering an open problem from [6], we prove that PrExt with fixed color bound k = 3 is NP-complete for bipartite (and even planar) graphs, and we prove a ge...

in this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. with this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. n(g) and s(g) are the set of all nilpotency classes and the set of all solvability classes for the group g with respect to different automorphi...