Ademir Hujdurović

Half-arc-transitive group actions with a small number of alternets

Abstract A graph X is said to be G-half-arc-transitive if G ⩽ Aut ( X ) acts transitively on the set of vertices of X and on the set of edges of X but does not act transitively on the set of arcs of X . Such graphs can be studied via corresponding alternets, that is, equivalence classes of the so-called reachability relation, first introduced by Cameron, Praeger and Wormald (1993) in [5] . If the vertex sets of two adjacent alternets either coincide or have half of their vertices in common the graph is said to be tightly attached . In this paper graphs admitting a half-arc-transitive group action with at most five alternets are considered. In particular, it is shown that if the number of alternets is at most three, then the graph is necessarily tightly attached, but there exist graphs with four and graphs with five alternets which are not tightly attached. The exceptional graphs all admit a partition giving the rose window graph R 6 ( 5 , 4 ) on 12 vertices as a quotient graph in case of four alternets, and a particular graph on 20 vertices in the case of five alternets.

Vertex-transitive CIS graphs

A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. A graph is well-covered if all its maximal stable sets are of the same size, co-well-covered if its complement is well-covered, and vertex-transitive if, for every pair of vertices, there exists an automorphism of the graph mapping one to the other. We show that a vertex-transitive graph is CIS if and only if it is well-covered, co-well-covered, and the product of its clique and stability numbers equals its order. A graph is irreducible if no two distinct vertices have the same neighborhood. We classify irreducible well-covered CIS graphs with clique number at most 3 and vertex-transitive CIS graphs of valency at most 7, which include an infinite family. We also exhibit an infinite family of vertex-transitive CIS graphs which are not Cayley.

Complexity and Algorithms for Finding a Perfect Phylogeny from Mixed Tumor Samples

Hajirasouliha and Raphael (WABI 2014) proposed a model for deconvoluting mixed tumor samples measured from a collection of high-throughput sequencing reads. This is related to understanding tumor evolution and critical cancer mutations. In short, their formulation asks to split each row of a binary matrix so that the resulting matrix corresponds to a perfect phylogeny and has the minimum number of rows among all matrices with this property. In this paper, we disprove several claims about this problem, including an NP-hardness proof of it. However, we show that the problem is indeed NP-hard, by providing a different proof. We also prove NP-completeness of a variant of this problem proposed in the same paper. On the positive side, we propose an efficient (though not necessarily optimal) heuristic algorithm based on coloring co-comparability graphs, and a polynomial time algorithm for solving the problem optimally on matrix instances in which no column is contained in both columns of a pair of conflicting columns. Implementations of these algorithms are freely available at https://github.com/alexandrutomescu/MixedPerfectPhylogeny.

On colour-preserving automorphisms of Cayley graphs

We study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups G , such that every such automorphism of every connected Cayley graph on G has a very simple form: the composition of a left-translation and a group automorphism. We find classes of groups that have the property, and we determine the orders of all groups that do not have the property. We also have analogous results for automorphisms that permute the colours, rather than preserving them.

Quasi m-Cayley circulants

A graph Γ is called a quasi m-Cayley graph on a group G if there exists a vertex ∞ ∈  V (Γ ) and a subgroup G of the vertex stabilizer Aut(Γ ) ∞ of the vertex ∞ in the full automorphism group Aut(Γ ) of Γ , such that G acts semiregularly on V (Γ ) ∖ {∞} with m orbits. If the vertex ∞ is adjacent to only one orbit of G on V (Γ ) ∖ {∞} , then Γ is called a strongly quasi m -Cayley graph on G . In this paper complete classifications of quasi 2 -Cayley, quasi 3 -Cayley and strongly quasi 4 -Cayley connected circulants are given.

Odd automorphisms in vertex-transitive graphs

An automorphism of a graph is said to be even / odd if it acts on the set of vertices as an even/odd permutation. In this article we pose the problem of determining which vertex-transitive graphs admit odd automorphisms. Partial results for certain classes of vertex-transitive graphs, in particular for Cayley graphs, are given. As a consequence, a characterization of arc-transitive circulants without odd automorphisms is obtained.

Finding a Perfect Phylogeny from Mixed Tumor Samples

Recently, Hajirasouliha and Raphael (WABI 2014) proposed a model for deconvoluting mixed tumor samples measured from a collection of high-throughput sequencing reads. This is related to understanding tumor evolution and critical cancer mutations. In short, their formulation asks to split each row of a binary matrix so that the resulting matrix corresponds to a perfect phylogeny and has the minimum number of rows among all matrices with this property. In this paper we disprove several claims about this problem, including an NP-hardness proof of it. However, we show that the problem is indeed NP-hard, by providing a different proof. We also prove NP-completeness of a variant of this problem proposed in the same paper. On the positive side, we obtain a polynomial time algorithm for matrix instances in which no column is contained in both columns of a pair of conflicting columns.

The Minimum Conflict-Free Row Split Problem Revisited

Motivated by applications in cancer genomics and following the work of Hajirasouliha and Raphael (WABI 2014), Hujdurovic et al. (IEEE TCBB, to appear) introduced the minimum conflict-free row split (MCRS) problem: split each row of a given binary matrix into a bitwise OR of a set of rows so that the resulting matrix corresponds to a perfect phylogeny and has the minimum number of rows among all matrices with this property. Hajirasouliha and Raphael also proposed the study of a similar problem, referred to as the minimum distinct conflict-free row split (MDCRS) problem, in which the task is to minimize the number of distinct rows of the resulting matrix. Hujdurovic et al. proved that both problems are NP-hard, gave a related characterization of transitively orientable graphs, and proposed a polynomial-time heuristic algorithm for the MCRS problem based on coloring cocomparability graphs.

Quasi-λ-distance-balanced graphs

Abstract A graph G is said to be quasi- λ -distance-balanced if for every pair of adjacent vertices u and v , the number of vertices that are closer to u than to v is λ times bigger (or λ times smaller) than the number of vertices that are closer to v than to u , for some positive rational number λ > 1 . This paper introduces the concept of quasi- λ -distance-balanced graphs, and gives some interesting examples and constructions. It is proved that every quasi- λ -distance-balanced graph is triangle-free. It is also proved that the only quasi- λ -distance-balanced graphs of diameter two are complete bipartite graphs. In addition, quasi- λ -distance-balanced Cartesian and lexicographic products of graphs are characterized. Connections between symmetry properties of graphs and the metric property of being quasi- λ -distance-balanced are investigated. Several open problems are posed.

On Bounds for the Product Irregularity Strength of Graphs

For a graph