David Tankus

Greedily constructing Hamiltonian paths, Hamiltonian cycles and maximum linear forests

There are various greedy schemas to construct a maximal path in a given input graph. Associated with each such schema is the family of graphs where it always results a path of maximum length, or a Hamiltonian path/cycle, if there exists one. Considerable amount of work has been carried out, regarding the characterization and recognition problems of such graphs. We present here a systematic listing of previous results of this type and fill up most of the remaining empty entries.

On the recognition of k -equistable graphs

A graph G=(V,E) is called equistable if there exist a positive integer t and a weight function

Weighted well-covered claw-free graphs

w:V \longrightarrow \mathbb{N}

On Related Edges in Well-Covered Graphs without Cycles of Length 4 and 6

such that S⊆V is a maximal stable set of G if and only if w(S)=t. The function w, if exists, is called an equistable function of G. No combinatorial characterization of equistable graphs is known, and the complexity status of recognizing equistable graphs is open. It is not even known whether recognizing equistable graphs is in NP. Let k be a positive integer. An equistable graph G=(V,E) is said to be k-equistable if it admits an equistable function which is bounded by k. For every constant k, we present a polynomial time algorithm which decides whether an input graph is k-equistable.

Well-covered graphs without cycles of lengths 4, 5 and 6

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input claw-free graph G, we present an O(n^6)algortihm, whose input is a claw-free graph G, and output is the vector space of weight functions w, for which G is w-well-covered. A graph G is equimatchable if all its maximal matchings are of the same cardinality. Assume that a weight function w is defined on the edges of G. Then G is w-equimatchable if all its maximal matchings are of the same weight. For every graph G, the set of weight functions w such that G is w-equimatchable is a vector space. We present an O(m*n^4 + n^5*log(n)) algorithm which receives an input graph G, and outputs the vector space of weight functions w such that G is w-equimatchable.

On relating edges in graphs without cycles of length 4

A graph is well-covered if every maximal independent set has the same cardinality. The recognition problem of well-covered graphs is known to be co-NPC. The complexity status of the problem is not known if the input is restricted to graphs with no cycles of length 4. We conjecture that the problem is polynomial if the input graph does not contain cycles of length 4 and 6, and prove several theorems supporting our conjecture.

Complexity Results for Generating Subgraphs

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input graph G without cycles of length 4, 5, and 6, we characterize polynomially the vector space of weight functions w for which G is w-well-covered. Let B be an induced complete bipartite subgraph of G on vertex sets of bipartition B_{X} and B_{Y}. Assume that there exists an independent set S such that both the union of S and B_{X} and the union of S and B_{Y} are maximal independent sets of G. Then B is a generating subgraph of G, and it produces the restriction w(B_{X})=w(B_{Y}). It is known that for every weight function w, if G is w-well-covered, then the above restriction is satisfied. In the special case, where B_{X}={x} and B_{Y}={y}, we say that xy is a relating edge. Recognizing relating edges and generating subgraphs is an NP-complete problem. However, we provide a polynomial algorithm for recognizing generating subgraphs of an input graph without cycles of length 5, 6 and 7. We also present a polynomial algorithm for recognizing relating edges in an input graph without cycles of length 5 and 6.

Well-dominated graphs without cycles of lengths 4 and 5

Abstract An edge xy is relating in the graph G if there is an independent set S, containing neither x nor y, such that S ∪ { x } and S ∪ { y } are both maximal independent sets in G. It is an NP-complete problem to decide whether an edge is relating [1] . We show that the problem remains NP-complete even for graphs without cycles of lengths 4 and 5. On the other hand, we show that for graphs without cycles of lengths 4 and 6, the problem can be solved in polynomial time.

Lower Bounds on the Odds Against Tree Spectral Sets

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space, denoted WCW(G). Let B be a complete bipartite induced subgraph of G on vertex sets of bipartition