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Papers by Elzbieta Sidorowicz

We denote by I the class of all finite simple graphs. A graph property is a nonempty isomorphism-... more We denote by I the class of all finite simple graphs. A graph property is a nonempty isomorphism-closed subclass of I. A property P is called (induced) hereditary if it is closed under (induced) subgraphs.

Discussiones Mathematicae Graph Theory

Fundamenta Informaticae

ABSTRACT

Discrete mathematics & theoretical computer science DMTCS

A k-colouring of a graph G is called acyclic if for every two distinct colours i and j, the subgr... more A k-colouring of a graph G is called acyclic if for every two distinct colours i and j, the subgraph induced in G by all the edges linking a vertex coloured with i and a vertex coloured with j is acyclic. In other words, there are no bichromatic alternating cycles. In 1999 Boiron et al. conjectured that a graph G with maximum degree at most 3 has an acyclic 2-colouring such that the set of vertices in each colour induces a subgraph with maximum degree at most 2. In this paper we prove this conjecture and show that such a colouring of a cubic graph can be determined in polynomial time. We also prove that it is an NP-complete problem to decide if a graph with maximum degree 4 has the above mentioned colouring.

The electronic journal of combinatorics

We consider the two-player game dened as follows. Let (G; L) be a graph G with a list assignment ... more We consider the two-player game dened as follows. Let (G; L) be a graph G with a list assignment L on its vertices. The two players, Alice and Bob, play alternately on G, Alice having the rst move. Alice's goal is to provide an L-colouring of G and Bob's goal is to prevent her from doing so. A move consists in choosing an uncoloured vertex v and assigning it a colour from the set L(v). Adjacent vertices of the same colour must not occur. This game will be called game list colouring. The game choice number of G, denoted by chg(G), is dened as the least k such that Alice has a winning strategy for any k-list assignment of G. We characterize the class of graphs with chg(G) 2 and determine the game choice number for some classes of graphs.

Discrete Applied Mathematics, 2014

ABSTRACT

Central European Journal of Mathematics, 2014

ABSTRACT

The Electronic Journal of Combinatorics, 2007

We consider the two-player game dened as follows. Let (G; L) be a graph G with a list assignment ... more We consider the two-player game dened as follows. Let (G; L) be a graph G with a list assignment L on its vertices. The two players, Alice and Bob, play alternately on G, Alice having the rst move. Alice's goal is to provide an L-colouring of G and Bob's goal is to prevent her from doing so. A move consists

Theoretical Computer Science, 2014

ABSTRACT A feedback vertex set of a graph G is a set S of its vertices such that the subgraph ind... more ABSTRACT A feedback vertex set of a graph G is a set S of its vertices such that the subgraph induced by V(G)∖SV(G)∖S is a forest. The cardinality of a minimum feedback vertex set of G is denoted by ∇(G)∇(G). A graph G is 2-degenerate if each subgraph G′G′ of G has a vertex v such that dG′(v)≤2dG′(v)≤2. In this paper, we prove that ∇(G)≤2n/5∇(G)≤2n/5 for any 2-degenerate n -vertex graph G and moreover, we show that this bound is tight. As a consequence, we derive a polynomial time algorithm, which for a given 2-degenerate n -vertex graph returns its feedback vertex set of cardinality at most 2n/52n/5.

Science China Mathematics, 2014

ABSTRACT

The Electronic Journal of Combinatorics - Electr. J. Comb., 2005

For graphs G;F and H we write G ! (F;H )t o mean that if the edges ofG are coloured with two colo... more For graphs G;F and H we write G ! (F;H )t o mean that if the edges ofG are coloured with two colours, say red and blue, then the red subgraph contains a copy of F or the blue subgraph contains a copy of H. The graph G is (F;H)-minimal (Ramsey-minimal )i fG ! (F;H) but G0 6! (F;H) for any proper subgraph G0 G. The class of all (F;H)-minimal graphs shall be denoted by R(F;H). In this paper we will determine the graphs in R(K1;2;K3).

Discussiones Mathematicae Graph Theory, 2009

Discussiones Mathematicae Graph Theory, 1997

Discussiones Mathematicae Graph Theory, 2002

Discussiones Mathematicae Graph Theory, 2006

ABSTRACT

Discrete Mathematics, 2007

We consider the version of a colouring game introduced by Bodlaender [On the complexity of some c... more We consider the version of a colouring game introduced by Bodlaender [On the complexity of some colorings games, Internat. J. Found. Comput. Sci. 2 (1991) 133-147]. We combine the concepts: this game and the generalised colouring of graphs as follows. The two players are Alice and Bob and they play alternatively with Alice having the first move. Let be given a graph G and an ordered set of hereditary properties (P 1 , P 2 , . . . , P n ). The players take turns colouring G with colours from {1, . . . , n} such that for each i = 1, 2, . . . , n the induced subgraph G[V i ] (V i is the set of vertices of G with colour i) has the property P i after each move of the players. If after |V (G)| moves the graph G is (P 1 , P 2 , . . . , P n )-partitioned (generalised coloured) then Alice wins. In this case, we say that the graph G has the property P 1 · · · P n . We characterise the class O O of graphs and we give an answer to a question, for k = 2, posed by Zhu [The game coloring number of planar graphs, J. Combin. Theory B 75 245-258]. We describe a new strategy for Alice for playing the (O O O 1 )-game on acyclic graphs. Also some open problems are posed.

Discrete Mathematics, 2002

Let P be a family of graphs. A graph G is said to satisfy a property P locally if G[N (v)] ∈ P fo... more Let P be a family of graphs. A graph G is said to satisfy a property P locally if G[N (v)] ∈ P for every v ∈ V (G). The class of graphs that satisÿes the property P locally will be denoted by L(P) and we shall call such a class a local property.

Discrete Mathematics, 2011

In this paper we will consider acyclic bipartition of the vertices of graphs, where acyclic means... more In this paper we will consider acyclic bipartition of the vertices of graphs, where acyclic means that the edges whose endpoints are in different parts of the partition induce a forest. We will require that the vertices belonging to the same partition induce graphs from particular class. We will search for acyclic bipartitions of cubic and subcubic graphs.

Discrete Mathematics, 2004

We denote by I the class of all finite simple graphs. A graph property is a nonempty isomorphism-... more We denote by I the class of all finite simple graphs. A graph property is a nonempty isomorphism-closed subclass of I. A property P is called (induced) hereditary if it is closed under (induced) subgraphs.

Discussiones Mathematicae Graph Theory

Fundamenta Informaticae

ABSTRACT

Discrete mathematics & theoretical computer science DMTCS

A k-colouring of a graph G is called acyclic if for every two distinct colours i and j, the subgr... more A k-colouring of a graph G is called acyclic if for every two distinct colours i and j, the subgraph induced in G by all the edges linking a vertex coloured with i and a vertex coloured with j is acyclic. In other words, there are no bichromatic alternating cycles. In 1999 Boiron et al. conjectured that a graph G with maximum degree at most 3 has an acyclic 2-colouring such that the set of vertices in each colour induces a subgraph with maximum degree at most 2. In this paper we prove this conjecture and show that such a colouring of a cubic graph can be determined in polynomial time. We also prove that it is an NP-complete problem to decide if a graph with maximum degree 4 has the above mentioned colouring.

The electronic journal of combinatorics

We consider the two-player game dened as follows. Let (G; L) be a graph G with a list assignment ... more We consider the two-player game dened as follows. Let (G; L) be a graph G with a list assignment L on its vertices. The two players, Alice and Bob, play alternately on G, Alice having the rst move. Alice's goal is to provide an L-colouring of G and Bob's goal is to prevent her from doing so. A move consists in choosing an uncoloured vertex v and assigning it a colour from the set L(v). Adjacent vertices of the same colour must not occur. This game will be called game list colouring. The game choice number of G, denoted by chg(G), is dened as the least k such that Alice has a winning strategy for any k-list assignment of G. We characterize the class of graphs with chg(G) 2 and determine the game choice number for some classes of graphs.

Discrete Applied Mathematics, 2014

ABSTRACT

Central European Journal of Mathematics, 2014

ABSTRACT

The Electronic Journal of Combinatorics, 2007

We consider the two-player game dened as follows. Let (G; L) be a graph G with a list assignment ... more We consider the two-player game dened as follows. Let (G; L) be a graph G with a list assignment L on its vertices. The two players, Alice and Bob, play alternately on G, Alice having the rst move. Alice's goal is to provide an L-colouring of G and Bob's goal is to prevent her from doing so. A move consists

Theoretical Computer Science, 2014

ABSTRACT A feedback vertex set of a graph G is a set S of its vertices such that the subgraph ind... more ABSTRACT A feedback vertex set of a graph G is a set S of its vertices such that the subgraph induced by V(G)∖SV(G)∖S is a forest. The cardinality of a minimum feedback vertex set of G is denoted by ∇(G)∇(G). A graph G is 2-degenerate if each subgraph G′G′ of G has a vertex v such that dG′(v)≤2dG′(v)≤2. In this paper, we prove that ∇(G)≤2n/5∇(G)≤2n/5 for any 2-degenerate n -vertex graph G and moreover, we show that this bound is tight. As a consequence, we derive a polynomial time algorithm, which for a given 2-degenerate n -vertex graph returns its feedback vertex set of cardinality at most 2n/52n/5.

Science China Mathematics, 2014

ABSTRACT

The Electronic Journal of Combinatorics - Electr. J. Comb., 2005

For graphs G;F and H we write G ! (F;H )t o mean that if the edges ofG are coloured with two colo... more For graphs G;F and H we write G ! (F;H )t o mean that if the edges ofG are coloured with two colours, say red and blue, then the red subgraph contains a copy of F or the blue subgraph contains a copy of H. The graph G is (F;H)-minimal (Ramsey-minimal )i fG ! (F;H) but G0 6! (F;H) for any proper subgraph G0 G. The class of all (F;H)-minimal graphs shall be denoted by R(F;H). In this paper we will determine the graphs in R(K1;2;K3).

Discussiones Mathematicae Graph Theory, 2009

Discussiones Mathematicae Graph Theory, 1997

Discussiones Mathematicae Graph Theory, 2002

Discussiones Mathematicae Graph Theory, 2006

ABSTRACT

Discrete Mathematics, 2007

We consider the version of a colouring game introduced by Bodlaender [On the complexity of some c... more We consider the version of a colouring game introduced by Bodlaender [On the complexity of some colorings games, Internat. J. Found. Comput. Sci. 2 (1991) 133-147]. We combine the concepts: this game and the generalised colouring of graphs as follows. The two players are Alice and Bob and they play alternatively with Alice having the first move. Let be given a graph G and an ordered set of hereditary properties (P 1 , P 2 , . . . , P n ). The players take turns colouring G with colours from {1, . . . , n} such that for each i = 1, 2, . . . , n the induced subgraph G[V i ] (V i is the set of vertices of G with colour i) has the property P i after each move of the players. If after |V (G)| moves the graph G is (P 1 , P 2 , . . . , P n )-partitioned (generalised coloured) then Alice wins. In this case, we say that the graph G has the property P 1 · · · P n . We characterise the class O O of graphs and we give an answer to a question, for k = 2, posed by Zhu [The game coloring number of planar graphs, J. Combin. Theory B 75 245-258]. We describe a new strategy for Alice for playing the (O O O 1 )-game on acyclic graphs. Also some open problems are posed.

Discrete Mathematics, 2002

Let P be a family of graphs. A graph G is said to satisfy a property P locally if G[N (v)] ∈ P fo... more Let P be a family of graphs. A graph G is said to satisfy a property P locally if G[N (v)] ∈ P for every v ∈ V (G). The class of graphs that satisÿes the property P locally will be denoted by L(P) and we shall call such a class a local property.

Discrete Mathematics, 2011

In this paper we will consider acyclic bipartition of the vertices of graphs, where acyclic means... more In this paper we will consider acyclic bipartition of the vertices of graphs, where acyclic means that the edges whose endpoints are in different parts of the partition induce a forest. We will require that the vertices belonging to the same partition induce graphs from particular class. We will search for acyclic bipartitions of cubic and subcubic graphs.

Discrete Mathematics, 2004