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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Colouring game and generalized colouring game on graphs with cut-vertices

100%
Sidorowicz E.
Discussiones Mathematicae Graph Theory
|
2010
|
tom 30
|
nr 3
499-533
EN
For k ≥ 2 we define a class of graphs 𝓗 ₖ = {G: every block of G has at most k vertices}. The class 𝓗 ₖ contains among other graphs forests, Husimi trees, line graphs of forests, cactus graphs. We consider the colouring game and the generalized colouring game on graphs from 𝓗 ₖ.
2
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Weakly P-saturated graphs

64%
Borowiecki M., Sidorowicz E.
Discussiones Mathematicae Graph Theory
|
2002
|
tom 22
|
nr 1
17-29
EN
For a hereditary property 𝓟 let $k_{𝓟}(G)$ denote the number of forbidden subgraphs contained in G. A graph G is said to be weakly 𝓟-saturated, if G has the property 𝓟 and there is a sequence of edges of G̅, say $e₁,e₂,...,e_l$, such that the chain of graphs $G = G₀ ⊂ G_0 + e₁ ⊂ G₁ + e₂ ⊂ ... ⊂ G_{l-1} + e_l = G_l = K_n(G_{i+1} = G_i + e_{i+1})$ has the following property: $k_{𝓟}(G_{i+1}) > k_{𝓟}(G_i)$, 0 ≤ i ≤ l-1. In this paper we shall investigate some properties of weakly saturated graphs. We will find upper bound for the minimum number of edges of weakly 𝓓ₖ-saturated graphs of order n. We shall determine the number wsat(n,𝓟) for some hereditary properties.
3
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Secure sets and their expansion in cubic graphs

64%
Jesse-Józefczyk K., Sidorowicz E.
Open Mathematics
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2014
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tom 12
|
nr 11
1664-1673
EN
Consider a graph whose vertices play the role of members of the opposing groups. The edge between two vertices means that these vertices may defend or attack each other. At one time, any attacker may attack only one vertex. Similarly, any defender fights for itself or helps exactly one of its neighbours. If we have a set of defenders that can repel any attack, then we say that the set is secure. Moreover, it is strong if it is also prepared for a raid of one additional foe who can strike anywhere. We show that almost any cubic graph of order n has a minimum strong secure set of cardinality less or equal to n/2 + 1. Moreover, we examine the possibility of an expansion of secure sets and strong secure sets.
4
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On the Non-(p−1)-Partite Kp-Free Graphs

51%
Amin K., Faudree J., Gould R. J., Sidorowicz E.
Discussiones Mathematicae Graph Theory
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2013
|
tom 33
|
nr 1
9-23
EN
We say that a graph G is maximal Kp-free if G does not contain Kp but if we add any new edge e ∈ E(G) to G, then the graph G + e contains Kp. We study the minimum and maximum size of non-(p − 1)-partite maximal Kp-free graphs with n vertices. We also answer the interpolation question: for which values of n and m are there any n-vertex maximal Kp-free graphs of size m?
5
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Generalized domination, independence and irredudance in graphs

51%
Borowiecki M., Michalak D., Sidorowicz E.
Discussiones Mathematicae Graph Theory
|
1997
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tom 17
|
nr 1
147-153
EN
The purpose of this paper is to present some basic properties of 𝓟-dominating, 𝓟-independent, and 𝓟-irredundant sets in graphs which generalize well-known properties of dominating, independent and irredundant sets, respectively.
6
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Extremal bipartite graphs with a unique k-factor

51%
Hoffmann A., Sidorowicz E., Volkmann L.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
|
nr 2
181-192
EN
Given integers p > k > 0, we consider the following problem of extremal graph theory: How many edges can a bipartite graph of order 2p have, if it contains a unique k-factor? We show that a labeling of the vertices in each part exists, such that at each vertex the indices of its neighbours in the factor are either all greater or all smaller than those of its neighbours in the graph without the factor. This enables us to prove that every bipartite graph with a unique k-factor and maximal size has exactly 2k vertices of degree k and 2k vertices of degree (|V(G)|)/2. As our main result we show that for k ≥ 1, p ≡ t mod k, 0 ≤ t < k, a bipartite graph G of order 2p with a unique k-factor meets 2|E(G)| ≤ p(p+k)-t(k-t). Furthermore, we present the structure of extremal graphs.
7
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Preface

51%
Drgas-Burchardt E., Sidorowicz E.
Discussiones Mathematicae Graph Theory
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2015
|
tom 35
|
nr 2
313-314
8
Content available remote

Preface

51%
Drgas-Burchardt E., Sidorowicz E.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 2
299-300
9
Content available remote

Preface

45%
Drgas-Burchardt E., Hałuszczak M., Kwaśnik M., Michalak D., Sidorowicz E.
Discussiones Mathematicae Graph Theory
|
2013
|
tom 33
|
nr 1
5-6
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