Christoph Brause

New sufficient conditions for α -redundant vertices

A vertex v in a graph G is called α -redundant if α ( G - v ) = α ( G ) , where α ( G ) stands for the independence number of G , i.e.?the maximum size of a subset of pairwise non-adjacent vertices. We will recall some results about α -redundant vertices and show some new sufficient conditions for a vertex to be α -redundant. Based on this, we will give a unified view about vertex removal techniques for solving the maximum independent set problem. It also leads to an efficient way to solve the problem in some subclasses of S 1 , 2 , 2 -free graphs and S 2 , 2 , 2 -free graphs, where S i , j , k is the graph consisting of three induced paths of lengths i , j and k with a common initial vertex.

The Maximum Independent Set Problem in Subclasses of Subcubic Graphs

The Maximum Independent Set problem is NP-hard and remains NP-hard for graphs of maximum degree at most three (also called subcubic graphs). In this paper, we will study its complexity in subclasses of subcubic graphs.Let S i , j , k be the graph consisting of three induced paths of lengths i , j and k , with a common initial vertex and A q l be the graph consisting of an induced cycle C q ( q ? 3 ) and an induced path with l edges having an end-vertex in common with the C q . For example S 0 , 0 , l and A 4 1 are known as path of length l and banner, respectively.Our main result is that the Maximum Independent Set problem can be solved in polynomial time in the class of subcubic, ( S 2 , j , k , A 5 l ) -free graphs.

A subexponential-time algorithm for the Maximum Independent Set Problem in Pt-free graphs

Abstract The Maximum Independent Set Problem is known to be NP-hard in general. In the last decades lots of effort were spent to find polynomial-time algorithms for P t -free graphs. A recent result presents such an algorithm for P 5 -free graphs. On the other hand, the complexity status for larger t is still unknown. In this paper we present a subexponential algorithm for the Maximum (Weighted) Independent Set Problem in P t -free graphs.

The Maximum Independent Set Problem in Subclasses of Si,j,k-Free Graphs

Abstract We consider the Maximum Independent Set (MIS) problem, which is known to be NP-hard in general, in subclasses of S i , j , k -free graphs, where S i , j , k is the graph consisting of three induced paths of lengths i , j , k with a common initial vertex. We revise the complexity of the problem. Then by two general approaches, augmenting graph and α-redundant vertex, some graph classes, for which we have polynomial solutions, are obtained.

Minimum Degree Conditions for the Proper Connection Number of Graphs

An edge-coloured graph G is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a graph G, denoted by pc(G), is the smallest number of colours that are needed in order to make G properly connected. In this paper, we consider sufficient conditions in terms of the ratio between minimum degree and order of a 2-connected graph G implying that G has proper connection number 2.

Extending the MAX Algorithm for Maximum Independent Set

Abstract The maximum independent set problem is an NP-hard problem. In this paper, we consider Algorithm MAX, which is a polynomial time algorithm for finding a maximal independent set in a graph G. We present a set of forbidden induced subgraphs such that Algorithm MAX always results in finding a maximum independent set of G. We also describe two modifications of Algorithm MAX and sets of forbidden induced subgraphs for the new algorithms.

Sum choice number of generalized θ-graphs

Abstract Let G = ( V E ) be a simple graph and for every vertex v ∈ V let L ( v ) be a set (list) of available colors. G is called L -colorable if there is a proper coloring φ of the vertices with φ ( v ) ∈ L ( v ) for all v ∈ V . A function f : V → N is called a choice function of G and G is said to be f -list colorable if G is L -colorable for every list assignment L choice function is defined by size ( f ) = ∑ v ∈ V f ( v ) and the sum choice number χ s c ( G ) denotes the minimum size of a choice function of G . Sum list colorings were introduced by Isaak in 2002 and got a lot of attention since then. For r ≥ 3 a generalized θ k 1 k 2 … k r -graph is a simple graph consisting of two vertices v 1 and v 2 connected by r internally vertex disjoint paths of lengths k 1 , k 2 , … , k r ( k 1 ≤ k 2 ≤ ⋯ ≤ k r ) . In 2014, Carraher et al. determined the sum-paintability of all generalized θ -graphs which is an online-version of the sum choice number and consequently an upper bound for it. In this paper we obtain sharp upper bounds for the sum choice number of all generalized θ -graphs with k 1 ≥ 2 and characterize all generalized θ -graphs G which attain the trivial upper bound | V ( G ) | + | E ( G ) | .

Proper connection and size of graphs

Abstract An edge-coloured graph G is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a connected graph G , denoted by pc ( G ) , is the smallest number of colours that are needed in order to make G properly connected. Our main result is the following: Let G be a connected graph of order n and k ≥ 2 . If | E ( G ) | ≥ n − k − 1 2 + k + 2 , then pc ( G ) ≤ k except when k = 2 and G ∈ { G 1 , G 2 } , where G 1 = K 1 ∨ ( 2 K 1 + K 2 ) and G 2 = K 1 ∨ ( K 1 + 2 K 2 ) .

On the minimum degree and the proper connection number of graphs

Abstract An edge-coloured graph G is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a graph G, denoted by pc(G), is the smallest number of colours that are needed in order to make G properly connected. In this paper we consider sufficient conditions in terms of the ratio between minimum degree and order of a 2-connected graph G implying that G has proper connection number 2.

Odd connection and odd vertex-connection of graphs

Abstract In the last years, connection concepts such as rainbow connection and proper connection appeared in graph theory and received a lot of attention. In this paper, we present a general concept of connection in graphs. As a particular case, we introduce the odd connection number and the odd vertex-connection number of a graph. Furthermore, we compute and study the odd connection number and the odd vertex-connection number of graphs of various graph classes.