We study, in three parts, degree sequences of k-families (or k-uniform hypergraphs) and shifted k-families. • The first part collects for the first time in one place, various implications such as Threshold ⇒ Uniquely Realizable ⇒ Degree-Maximal ⇒ Shifted which are equivalent concepts for 2-families (= simple graphs), but strict implications for kfamilies with k ≥ 3. The implication that uniquel...

We find the maximum number of maximal independent sets in two families of graphs: all graphs with n vertices and at most r cycles, and all such graphs that are also connected. In addition, we characterize the extremal graphs. This proves a strengthening of a conjecture of Goh and Koh [3]. We do the same for the maximum number of maximum independent sets, generalizing a theorem of Jou and Chang ...

It was recently proved that any graph satisfying ω > 2 3(∆ + 1) contains a stable set hitting every maximum clique. In this note we prove that the same is true for graphs satisfying ω ≥ 23(∆ + 1) unless the graph is the strong product of an odd hole and Kω/2. We also provide a counterexample to a recent conjecture on the existence of a stable set hitting every sufficiently large maximal clique.

We review the Green/Kleitman/Leeb interpretation of de Bruijn's symmetric chain decomposition of Nn, and explain how it can be used to find a maximal collection of disjoint symmetric chains in the nonsymmetric lattice of partitions of a set.

A graph G is called CIS if each maximal clique intersects each maximal stable set in G, and is called almost CIS if it has a unique disjoint pair (C, S) consisting of a maximal clique C and a maximal stable set S. While it is still unknown if there exists a good structural characterization of all CIS graphs, in this note we prove the following Andrade-BorosGurvich conjecture: A graph is almost ...

A path of a graph is maximal if it is not a proper subpath of any other path of the graph. A graph is scenic if every maximal path of the graph is a maximum length path. In [4] we give a new proof of C. Thomassen's result characterizing all scenic graphs with Hamiltonian path. Using similar methods here we determine all scenic graphs with no Hamiltonian path.

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 o...

A triangle-free graph is maximal if the addition of any edge creates a triangle. For n ~> 5, we show there is an n-node m-edge maximal triangle-free graph if and only if it is complete bipartite or 2n-5<<.m<<.L(n-1)2/4J+l. A diameter 2 graph is minimal if the deletion of any edge increases the diameter. We show that a triangle-free graph is maximal if and only if it is minimal of diameter 2. Fo...

In this paper, within the framework of process mining we examine Maximal Pattern Mining method introduced by Liesaputra et al. in [1]. This constructs a transition graph, i.e. labelled directed graph for traces with similar structure. The idea behind algorithm is to analyze event log, identify loops, parallel events and optionality between them, order determine maximal patterns. [1], authors pr...

1.1 Maximum Clique Problem Assume that the finite undirected simple graph is given, where V is the set of nodes, E is the set of edges. The arbitrary full graph is called a clique. The clique, which does not contain other cliques, is called a maximal Clique. The largest maximal clique is called a maximum clique. To extract all maximal cliques from the graph G. Many algorithms have...