Maximum Cardinality Research Articles

Distance k-Cost Effective Sets in the Corona and Lexicographic Product of Graphs

Let G be a connected graph and k ≥ 1 be an integer. The open k-neighborhood Nk G(v) of v ∈ V (G) is the set Nk G(v) = {u ∈ V (G) \ {v}: dG(u, v) ≤ k}. A set S of vertices of G is called distance k-cost effective of G if for every vertex u in S, |Nk G(u) ∩ (V (G) \ S)| − |Nk G(u) ∩ S| ≥ 0. The maximum cardinality of a distance k-cost effective set of G is called the upper distance k-cost effective number of G. In this paper, we characterized the distance k-cost effective sets in the corona and lexicographic product of two graphs. Consequently, the bounds or the exact values of the upper distance k-cost effective numbers of these graphs are obtained.

Cliques and Supercliques in Graphs

A set S ⊆ V (G) of an undirected graph G is a clique if every two distinct vertices in S are adjacent. A clique is a superclique if for every pair of distinct vertices v, w ∈ S, there exists u ∈ V (G) \ S such that u ∈ NG(v) \ NG(w) or u ∈ NG(w) \ NG(v). The maximum cardinality of a clique (resp. superclique) in G is called the clique (resp. superclique) number of G. In this paper, we determine the clique and superclique numbers of some graphs.

New Results on k-independence of Hypergraphs

Let $H=(V,E)$ be an $s$-uniform hypergraph of order $n$ and $k\geq 0$ be an integer. A $k$-independent set $S\subseteq H$ is a set of vertices such that the maximum degree in the hypergraph induced by $S$ is at most $k$. Denoted by $\alpha_k(H)$ the maximum cardinality of the $k$-independent set of $H$. In this paper, we first give a lower bound of $\alpha_k(H)$ by the maximum degree of $H$. Furthermore, we prove that $\alpha_k(H)\geq \frac{s(k+1)n}{2d+s(k+1)}$ where $d$ is average degree of $H$, and $k\geq 0$ is an integer.

The Upper Connected Outer Connected Monophonic Number of a Graph

For a connected graph of order at least two, a connected outer connected monophonic set of is called a minimal connected outer connected monophonic set if no proper subset of is a connected outer connected monophonic set of . The upper connected outer connected monophonic number of is the maximum cardinality of a minimal connected outer connected monophonic set of . We determine bounds for it and find the upper connected outer connected monophonic number of certain classes of graphs. It is shown that for any two integers with , there is a connected graph of order with and . Also, for any three integers and with , there is a connected graph with and and a minimal connected outer connected monophonic set of cardinality , where is the connected outer connected monophonic number of a graph.

On clique numbers of colored mixed graphs

An (m,n)-colored mixed graph, or simply, an (m,n)-graph is a graph having m different types of arcs and n different types of edges. A homomorphism of an (m,n)-graph G to another (m,n)-graph H is a vertex mapping that preserves adjacency, the type thereto and the direction. A subset R of the set of vertices of G that always maps distinct vertices in itself to distinct image vertices under any homomorphism is called an (m,n)-relative clique of G. The maximum cardinality of an (m,n)-relative clique of a graph is called the (m,n)-relative clique number of the graph. In this article, we explore the (m,n)-relative clique numbers for various families of graphs.

On b-acyclic chromatic number of a graph

Let G be a graph. We introduce the acyclic b-chromatic number of G as an analogue to the b-chromatic number of G. An acyclic coloring of a graph G is a map c:V(G)rightarrow {1,ldots ,k} such that c(u)ne c(v) for any uvin E(G) and the induced subgraph on vertices of any two colors i,jin {1,ldots ,k} induces a forest. On the set of all acyclic colorings of G we define a relation whose transitive closure is a strict partial order. The minimum cardinality of its minimal element is then the acyclic chromatic number A(G) of G and the maximum cardinality of its minimal element is the acyclic b-chromatic number A_{textrm{b}}(G) of G. We present several properties of A_{textrm{b}}(G). In particular, we derive A_{textrm{b}}(G) for several known graph families, derive some bounds for A_{textrm{b}}(G), compare A_{textrm{b}}(G) with some other parameters and generalize some influential tools from b-colorings to acyclic b-colorings.

Open packing subdivision number of graphs

A nonempty set [Formula: see text] of a graph [Formula: see text] is an open packing set of [Formula: see text] if no two vertices of [Formula: see text] have a common neighbor in [Formula: see text]. The maximum cardinality of an open packing set is called the open packing number of [Formula: see text] and is denoted by [Formula: see text]. The open packing subdivision number [Formula: see text] is the minimum number of edges in [Formula: see text] that must be subdivided (each edge in [Formula: see text] can be subdivided at most once) in order to increase the open packing number. In this paper, we initiate a study on this parameter.

Acyclic matching in some subclasses of graphs

A subset M⊆E of edges of a graph G=(V,E) is called a matching if no two edges of M share a common vertex. Given a matching M in G, a vertex v∈V is called M-saturated if there exists an edge e∈M incident with v. A matching M of a graph G is called an acyclic matching if, G[V(M)], the subgraph of G induced by the M-saturated vertices of G is an acyclic graph. Given a graph G, the Acyclic Matching problem asks to find an acyclic matching of maximum cardinality in G. The Decide-Acyclic Matching problem takes a graph G and an integer k and asks whether G has an acyclic matching of cardinality at least k. The Decide-Acyclic Matching problem is known to be NP-complete for general graphs as well as for bipartite graphs. In this paper, we strengthen this result by showing that the Decide-Acyclic Matching problem remains NP-complete for comb-convex bipartite graphs, star-convex bipartite graphs, and dually chordal graphs. On the positive side, we show that the Acyclic Matching problem is linear time solvable for split graphs, block graphs, and proper interval graphs. We show that the Acyclic Matching problem is hard to approximate within a factor of n1−ϵ for any ϵ>0 unless P=NP. Also, we show that the Acyclic Matching problem is APX-complete for (2k+1)-regular graphs for every fixed integer k≥3.

Irredundance graphs

A set D of vertices of a graph G = ( V , E ) is irredundant if each v ∈ D satisfies (a) v is isolated in the subgraph induced by D , or (b) v is adjacent to a vertex in V − D that is nonadjacent to all other vertices in D . The upper irredundance number IR ( G ) is the largest cardinality of an irredundant set of G ; an IR ( G ) -set is an irredundant set of cardinality IR ( G ) . The IR -graph of G has the irredundant sets of G of maximum cardinality, that is, the IR ( G ) -sets, as vertex set, and sets D and D ′ are adjacent if and only if D ′ is obtained from D by exchanging a single vertex of D for an adjacent vertex in D ′ . We study the realizability of graphs as IR -graphs and show that all disconnected graphs are IR -graphs, but some connected graphs (e.g. stars K 1 , n , n ≥ 2 , P 4 , P 5 , C 5 , C 6 , C 7 ) are not. We show that the double star S ( 2 , 2 ) – the tree obtained by joining the two central vertices of two disjoint copies of P 3 – is the unique smallest IR -tree with diameter 3 and also a smallest non-complete IR -tree, and the tree obtained by subdividing a single pendant edge of S ( 2 , 2 ) is the unique smallest IR -tree with diameter 4.

Induced subgraphs of a tree with constraint degree

For an integer k≥2, a vertex partition (V1,…,Vs) of a graph G is called a k-good partition if dG[Vi](v)≡1 (mod k) for each v∈Vi, i∈{1,…,s}. We characterize all trees with a k-good partition. Let f1,k(G)=max{|V(H)|: H is an induced subgraph H of G with dH(v)≡1 (mod k) for every vertex v}. In 1997, Berman et al. (1997) showed that f1,k(T)≥2⌊n+2k−32k−1⌋ for any tree T of order n. By sacrificing the bound slightly, but with a much simpler way, we are able to show that f1,k(T)≥nk, with equality if and only if T is the balanced double star of order 2k. Let f0,k1(T) be the maximum cardinality of a subset S⊆V(T) with dT[S](v)=1 or dT[S](v)≡0 (mod k) for each v∈S. In addition, we give a short proof of the result, due to Huang and Hou [9], that f0,k1(T)≥2n3 for any tree T and integer k≥3.

Twin-free cliques in annihilator graphs of commutative rings

For a connected graph [Formula: see text] a clique [Formula: see text] is twin-free if every pair of elements of [Formula: see text] have distinct closed neighborhoods and the number of elements in a twin-free clique of maximum cardinality is called twin-free clique number of [Formula: see text]. The annihilator graph [Formula: see text] of a commutative and unital ring [Formula: see text] is a graph whose vertices are all non-zero zero-divisors of [Formula: see text] and there is an edge between two distinct vertices [Formula: see text] if and only if [Formula: see text] is properly contained in [Formula: see text]. In this paper, twin-free clique number of [Formula: see text] is computed and as an application the strong metric dimension of [Formula: see text] is characterized. Among other things, for a reduced ring [Formula: see text], the forcing strong metric dimension of [Formula: see text] is given.

Halin’s end degree conjecture

An end of a graph G is an equivalence class of rays, where two rays are equivalent if there are infinitely many vertex-disjoint paths between them in G. The degree of an end is the maximum cardinality of a collection of pairwise disjoint rays in this equivalence class. Halin conjectured that the end degree can be characterised in terms of certain typical ray configurations, which would generalise his famous grid theorem. In particular, every end of regular uncountable degree κ would contain a star of rays, i.e., a configuration consisting of a central ray R and κ neighbouring rays (Ri: i < κ) all disjoint from each other and each Ri sending a family of infinitely many disjoint paths to R so that paths from distinct families only meet in R. We show that Halin’s conjecture fails for end degree 1א, holds for $${\aleph _2},{\aleph _3}, \ldots ,{\aleph _\omega}$$ fails for אω+1, and is undecidable (in ZFC) for the next אω+n with n ∈ ℕ, n ≽ 2. Further results include a complete solution for all cardinals under GCH, complemented by a number of consistency results.

Degree and Regularity of Eulerian Ideals of Hypergraphs

We define the Eulerian ideal of a $k$-uniform hypergraph and study its degree and Castelnuovo-Mumford regularity. The main tool is a Gröbner basis of the ideal obtained combinatorially from the hypergraph. We define the notion of parity join in a hypergraph and show that the regularity of the Eulerian ideal is equal to the maximum cardinality of such a set of edges. The formula for the degree involves the cardinality of the set of sets of vertices, $T$, that admit a $T$-join. We compute the degree and regularity explicitly in the cases of a complete $k$-partite hypergraph and a complete hypergraph of rank three.

On the Δ-interval and the Δ-convexity numbers of graphs and graph products

Given a graph G and a set S ⊆ V ( G ) , the Δ -interval of S , [ S ] Δ , is the set formed by the vertices of S and every w ∈ V ( G ) forming a triangle with two vertices of S . If [ S ] Δ = S , then S is Δ -convex of G ; if [ S ] Δ = V ( G ) , then S is a Δ -interval set of G . The Δ -interval number of G is the minimum cardinality of a Δ -interval set and the Δ -convexity number of G is the maximum cardinality of a proper Δ -convex subset of V ( G ) . In this work, we show that the problem of computing the Δ -convexity number is W[1]-hard and NP-hard to approximate within a factor O ( n 1 − ɛ ) for any constant ɛ > 0 even for graphs with diameter 2 and that the problem of computing the Δ -interval number is NP-complete for general graphs. For the positive side, we present characterizations that lead to polynomial-time algorithms for computing the Δ -convexity number of chordal graphs and for computing the Δ -interval number of block graphs. We also present results on the Δ -hull, Δ -interval and Δ -convexity numbers concerning the three standard graph products, namely, the Cartesian, the strong and the lexicographic products, in function of these and well-studied parameters of the operands.

Novel integer programming models for the stable kidney exchange problem

Kidney exchange programs (KEPs) represent an additional possibility of transplant for patients suffering from end-stage kidney disease. If a patient has a willing living donor with whom the patient is not compatible, the pair recipient–donor can join a pool of incompatible pairs and, if compatibility between recipient and donor in two or more pairs exists, organs can be exchanged between them. The problem can be modelled as an integer program that in general aims at finding the pairs that should be selected for transplant such that maximum number of transplants is performed.In this paper, we consider that for each recipient there may exist a preference order over the organs that he/she can receive, since a recipient may be compatible with several donors but the level of compatibility with the recipient might vary for different donors. Under this setting, the aim is to find the maximum cardinality stable exchange, a solution where no blocking cycle exists, i.e., there is no cycle such that all recipients prefer the donor in that cycle rather than that in the exchange. For this purpose we propose four novel integer programming models based on the well-known edge and cycle formulations, and also on the position-indexed formulation. These formulations are adjusted for both finding stable and strongly stable exchanges under strict preferences and for the case when ties in preferences may exist. Furthermore, we study a situation when the stability requirement can be relaxed by addressing the trade-off between maximum cardinality versus number of blocking cycles allowed in a solution. The effectiveness of the proposed models is assessed through extensive computational experiments on a wide set of instances. Results show that the cycle-edge and position-indexed formulations outperform the other two formulations. Another important practical outcome is that targeting strongly stable solutions has a much higher negative impact on the number of transplants (with an average reduction of up to 20% for the bigger instances), when compared to stable solutions.

Domination, Independence and Fibonacci Numbers in Graphs Containing Disjoint Cycles

Graph theory is a delightful playground for the exploration of proof techniques in discrete mathematics and its results have applications in many areas of the computing, social, and natural sciences. The fastest growing area within graph theory is the study of domination and Independence numbers. Domination number is the cardinality of a minimum dominating set of a graph. Independence number is the maximal cardinality of an independent set of vertices of a graph. The concept of Fibonacci numbers of graphs was first introduced by Prodinger and Tichy in 1982. The Fibonacci numbers of a graph is the number of independent vertex subsets. In this paper, introduce the identities of domination, independence and Fibonacci numbers of graphs containing vertex-disjoint cycles and edge-disjoint cycles.

Edge open packing sets in graphs

In a graphG = (V, E), two edgese1ande2are said to have acommon edgeif there exists an edgee ∈ E(G) different frome1ande2such thatejoins a vertex ofe1to a vertex ofe2inG. That is, 〈e1, e, e2〉 is eitherP4orK3inG. A non-empty setD ⊆ E(G) is anedge open packing setof a graphGif no two edges ofDhave a common edge inG. The maximum cardinality of an edge open packing set is theedge open packing numberofGand is denoted byρoe(G). In this paper, we initiate a study on this parameter.

Minimum cost b-matching problems with neighborhoods

In this paper, we deal with minimum cost b-matching problems on graphs where the nodes are assumed to belong to non-necessarily convex regions called neighborhoods, and the costs are given by the distances between points of the neighborhoods. The goal in the proposed problems is twofold: (i) finding a b-matching in the graph and (ii) determining a point in each neighborhood to be the connection point among the edges defining the b-matching. Different variants of the minimum cost b-matching problem are considered depending on the criteria to match neighborhoods: perfect, maximum cardinality, maximal and the a–b-matching problems. The theoretical complexity of solving each one of these problems is analyzed. Different mixed integer non-linear programming formulations are proposed for each one of the considered problems and then reformulated as Second Order Cone formulations. An extensive computational experience shows the efficiency of the proposed formulations to solve the problems under study.

Eulerian ideals

Let G be a simple graph and , where is the homomorphism that sends an edge to the product of its vertices. The ideal is Cohen–Macaulay, one-dimensional and binomial. If G is bipartite, it is known that the Castelnuovo–Mumford regularity of is equal to the maximum cardinality of a set of edges having no more than half of the edges of any Eulerian subgraph of G. Here, with respect to the grevlex order associated to an ordering of the edge set of G, we describe a Gröbner basis for , and we characterize the standard monomials of the ideal in terms of even sets of vertices marked with a parity. Using these results, we give a combinatorial interpretation of the degree of , via the set of even sets of vertices of G; and we show that the Castelnuovo–Mumford regularity of , for any graph, is the maximum cardinality of a set of edges having no more than half of the edges of any even Eulerian subgraph of G or, equivalently, the maximum cardinality of a minimum fixed parity T-join.

Supercliques in a Graph

A set S ⊆ V (G) of a (simple) undirected graph G is a superclique in G if it is a clique and for every pair of distinct vertices v, w ∈ S, there exists u ∈ V (G) \ S such that u ∈ NG(v) \ NG(w) or u ∈ NG(w) \ NG(v). The maximum cardinality among the supercliques in G, denoted by ωs(G), is called the superclique number of G. In this paper, we determine the superclique numbers of some graphs including those resulting from some binary operations of graphs. We will also show that the difference of the clique number and the superclique number can be made arbitrarily large.