Irmina A. Zioło

On-line arbitrarily vertex decomposable suns

We give a complete characterization of on-line arbitrarily vertex decomposable graphs in the family of unicycle graphs called suns. A sun is a graph with maximum degree three, such that deleting vertices of degree one results in a cycle. This result has already been used in another paper to prove some Ore-type conditions for on-line arbitrarily decomposable graphs.

Fixed-point-free embeddings of digraphs with small size

We prove that every digraph of order n and size at most 74n-81 is embeddable in its complement. Moreover, for such digraphs there are embeddings without fixed points.

Embedding digraphs of small size

Let D=(V,A) be a digraph without loops and multiple arcs of order n ⩾ 2. We say that D is embeddable if there is a permutation φ : V → V (called complementing permutation) such that (φ(x), φ(y)) ∉ A for every arc (x, y) ϵ A. In this paper we prove that if | A | < [3(n - 2)/2] then D is embeddable and, moreover, there is a complementing permutation which is a composition of disjoint transpositions when n is even or a composition of disjoint transpositions and a fixed point when n is odd.

On arbitraly vertex decomposable unicyclic graphs with dominating cycle

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n1, . . . , nk) of positive integers such that ∑k i=1 ni = n, there exists a partition (V1, . . . , Vk) of vertex set of G such that for every i ∈ {1, . . . , k} the set Vi induces a connected subgraph of G on ni vertices. We consider arbitrarily vertex decomposable unicyclic graphs with dominating cycle. We also characterize all such graphs with at most four hanging vertices such that exactly two of them have a common neighbour.

Subforests of bipartite digraphs — the minimum degree condition

Abstract Let D be a bipartite digraph and let F be an oriented forest of size k −1. We consider two conditions on the minimum indegree and the minimum outdegree of the digraph D guaranteeing that D contains F. These conditions extend older results concerning oriented trees of size k −1.

Steiner almost self-complementary graphs and halving near-Steiner triple systems

We show that for every admissible order v=0 or 2(mod6) there exists a near-Steiner triple system of order v that can be halved. As a corollary we obtain that a Steiner almost self-complementary graph with n vertices exists if and only if n=0 or 2(mod6).

Some families of arbitrarily vertex decomposable graphs

Abstract We give a complete characterization of some families of arbitrarily vertex decomposable unicyclic graphs. We consider also the ‘on-line’ version of the problem.