Finding Skew Partitions Efficiently

A skew partition as defined by Chvátal is a partition of the vertex set of a graph into four nonempty parts A, B, C, D such that there are all possible edges between A and B and no edges between C and D. We present a polynomial-time algorithm for testing whether a graph admits a skew partition. Our algorithm solves the more general list skew partition problem, where the input contains, for each vertex, a list containing some of the labels A, B, C, D of the four parts. Our polynomial-time algorithm settles the complexity of the original partition problem proposed by Chvátal in 1985 and answers a recent question of Feder, Hell, Klein, and Motwani.     

The minimum genus of a two-point universal graph

A graph G is two-point universal if, given any two vertices A and B, there is a vertex jointed to both, a vertex joined to neither, a vertex joined to A but not B, and a vertex joined to B but not A. Erdös asked whether there is an infinite family of such graphs of some genus γ. In this note, we show that the number of vertices of a two-point universal graph of genus γ satisfies n ≤ 216(2γ + 1)² so that there are most finitely many of each genus.     

On the extendability of Bi-Cayley graphs of finite abelian groups

Let G be a finite group and A a nonempty subset (possibly containing the identity element) of G. The Bi-Cayley graph X=BC(G,A) of G with respect to A is defined as the bipartite graph with vertex set G×{0,1} and edge set {{(g,0),(sg,1)}∣g∈G,s∈A}. A graph Γ admitting a perfect matching is called n-extendable if ∣V(Γ)∣≥2n+2 and every matching of size n in Γ can be extended to a perfect matching of Γ. In this paper, the extendability of Bi-Cayley graphs of finite abelian groups is explored. In particular, 2-extendable and 3-extendable Bi-Cayley graphs of finite abelian groups are characterized.     

Graphs Without Large Apples and the Maximum Weight Independent Set Problem

An apple A _k is the graph obtained from a chordless cycle C _k of length k ≥ 4 by adding a vertex that has exactly one neighbor on the cycle. The class of apple-free graphs is a common generalization of claw-free graphs and chordal graphs, two classes enjoying many attractive properties, including polynomial-time solvability of the maximum weight independent set problem. Recently, Brandstädt et al. showed that this property extends to the class of apple-free graphs. In the present paper, we study further generalization of this class called graphs without large apples: these are (A _k , A _k+1, . . .)-free graphs for values of k strictly greater than 4. The complexity of the maximum weight independent set problem is unknown even for k = 5. By exploring the structure of graphs without large apples, we discover a sufficient condition for claw-freeness of such graphs. We show that the condition is satisfied by bounded-degree and apex-minor-free graphs of sufficiently large tree-width. This implies an efficient solution to the maximum weight independent set problem for those graphs without large apples, which either have bounded vertex degree or exclude a fixed apex graph as a minor.     

Strongly regular graphs with Hoffman’s condition

It is known that if the minimal eigenvalue of a graph is ?2, then the graph satisfies Hoffman’s condition; i.e., for any generated complete bipartite subgraph K _1,3 with parts {p} and {q ₁, q ₂, q ₃}, any vertex distinct from p and adjacent to two vertices from the second part is not adjacent to the third vertex and is adjacent to p. We prove the converse statement, formulated for strongly regular graphs containing a 3-claw and satisfying the condition gm > 1.     

Some isomorphisms between pairs of latin squares

Let A, B, C, D be latin squares with A orthogonal to B and C orthogonal to D. The pair A, B is isomorphic with the pair C, D if the graph of A, B is graph-isomorphic with the graph of C, D. A characterization is given for determining when a pair A, B of latin squares is isomorphic with a self-orthogonal square C and its transpose. Self-orthogonal squares are important because they are both abundant and easy to store. An algorithm either displays a self-orthogonal square C and an isomorphism from A, B to C, C^T or, if none exists, gives a small set of blocks to the existence of such a square isomorphism.     

Two-factors in orientated graphs with forbidden transitions

The instance of the problem of finding 2-factors in an orientated graph with forbidden transitions consists of an orientated graph G and for each vertex v of G, a graph H_v of allowed transitions at v. Vertices of the graph H_v are the edges incident to v. An orientated 2-factor F of G is called legal if all the transitions are allowed, i.e. for every vertex v, the two edges of F adjacent to it form an edge in H_v. Deciding whether a legal 2-factor exists in G is NP-complete in general. We investigate the case when the graphs of allowed transitions are taken from some fixed class C. We provide an exact characterization of classes C so that the problem is NP-complete. In particular, we prove a dichotomy for this problem, i.e. that for any class C it is either polynomial or NP-complete.     

Connected graphs with maximal Q-index: The one-dominating-vertex case

By the signless Laplacian of a (simple) graph G we mean the matrix Q(G)=D(G)+A(G), where A(G),D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. For every pair of positive integers n,k, it is proved that if 3?k?n-3, then H_n,k, the graph obtained from the star K_1,n-1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the largest signless Laplacian eigenvalue over all connected graphs with n vertices and n+k edges.     

The satisfactory partition problem

The SATISFACTORY PARTITION problem consists in deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part. This problem was introduced by Gerber and Kobler [Partitioning a graph to satisfy all vertices, Technical report, Swiss Federal Institute of Technology, Lausanne, 1998; Algorithmic approach to the satisfactory graph partitioning problem, European J. Oper. Res. 125 (2000) 283-291] and further studied by other authors but its complexity remained open until now. We prove in this paper that SATISFACTORY PARTITION, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. However, for graphs with maximum degree at most 4 the problem is polynomially solvable. We also study generalizations and variants of this problem where a partition into k nonempty parts (k?3) is requested.     

Hadwiger numbers and over-dominating colourings

Motivated by Hadwiger’s conjecture, we say that a colouring of a graph is over-dominating if every vertex is joined to a vertex of each other colour and if, for each pair of colour classes C₁ and C₂, either C₁ has a vertex adjacent to all vertices in C₂ or C₂ has a vertex adjacent to all vertices in C₁.We show that a graph that has an over-dominating colouring with k colours has a complete minor of order at least 2k/3 and that this bound is essentially best possible.     

The graphs with only self-dual signings

Given a graph G, it is possible to attach positive and negative signs to its lines only, to its points only, or to both. The resulting structures are called respectively signed graphs, marked graphs and nets. The dual of each such structure is obtained by changing every sign in it. We determine all graphs G for which every suitable marked graph on G is self-dual (the M-dual graphs), and also the corresponding graphs G for signed graphs (S-dual) and for nets (N-dual.A graph G is M-dual if and only if G or ? is one of the graphs K_2m, 2K_m, mK₂, K_m + K₂ or 2C₄. The S-dual graphs are C₆, 2C₃, 2C₄, 2K_1n, 2nK₂, K_1,2n, nK_1,2, K_2n, K?_n and all graphs obtained from these by the addition of isolated points. Finally, the only N-dual graph other than -K_2n is 2K₂.     

The A4‐structure of a graph

We define the A₄‐structure of a graph G to be the 4‐uniform hypergraph on the vertex set of G whose edges are the vertex subsets inducing 2K₂, C₄, or P₄. We show that perfection of a graph is determined by its A₄‐structure. We relate the A₄‐structure to the canonical decomposition of a graph as defined by Tyshkevich [Discrete Math 220 (2000) 201–238]; for example, a graph is indecomposable if and only if its A₄‐structure is connected. We also characterize the graphs having the same A₄‐structure as a split graph.     

Characterization of k-variegated graphs, k ⩾ 3

A graph is said to be k-variegated if its vertex set can be partitioned into k equal parts such that each vertex is adjacent to exactly one vertex from every other part not containing it. Bednarek and Sanders [1] posed the problem of characterizing k-variegated graphs. V.N. Bhat-Nayak, S.A. Choudum and R.N. Naik [2] gave the characterization of 2-variegated graphs. In this paper we characterize k-variegated graphs for k ? 3.     

Game-perfect graphs

A graph coloring game introduced by Bodlaender (Int J Found Comput Sci 2:133–147, 1991) as coloring construction game is the following. Two players, Alice and Bob, alternately color vertices of a given graph G with a color from a given color set C, so that adjacent vertices receive distinct colors. Alice has the first move. The game ends if no move is possible any more. Alice wins if every vertex of G is colored at the end, otherwise Bob wins. We consider two variants of Bodlaender’s graph coloring game: one (A) in which Alice has the right to have the first move and to miss a turn, the other (B) in which Bob has these rights. These games define the A-game chromatic number resp. the B-game chromatic number of a graph. For such a variant g, a graph G is g-perfect if, for every induced subgraph H of G, the clique number of H equals the g-game chromatic number of H. We determine those graphs for which the game chromatic numbers are 2 and prove that the triangle-free B-perfect graphs are exactly the forests of stars, and the triangle-free A-perfect graphs are exactly the graphs each component of which is a complete bipartite graph or a complete bipartite graph minus one edge or a singleton. From these results we may easily derive the set of triangle-free game-perfect graphs with respect to Bodlaender’s original game. We also determine the B-perfect graphs with clique number 3. As a general result we prove that complements of bipartite graphs are A-perfect.      

Dimension-2 poset competition numbers and dimension-2 poset double competition numbers

Let D=(V(D),A(D)) be a digraph. The competition graph of D, is the graph with vertex set V(D) and edge set . The double competition graph of D, is the graph with vertex set V(D) and edge set . A poset of dimension at most two is a digraph whose vertices are some points in the Euclidean plane R² and there is an arc going from a vertex (x₁,y₁) to a vertex (x₂,y₂) if and only if x₁>x₂ and y₁>y₂. We show that a graph is the competition graph of a poset of dimension at most two if and only if it is an interval graph, at least half of whose maximal cliques are isolated vertices. This answers an open question on the doubly partial order competition number posed by Cho and Kim. We prove that the double competition graph of a poset of dimension at most two must be a trapezoid graph, generalizing a result of Kim, Kim, and Rho. Some connections are also established between the minimum numbers of isolated vertices required to be added to change a given graph into the competition graph, the double competition graph, of a poset and the minimum sizes of certain intersection representations of that graph.     

Total domination in claw-free graphs with minimum degree 2

A set S of vertices in a graph G is a total dominating set (TDS) of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a TDS of G is the total domination number of G, denoted by γ_t(G). A graph is claw-free if it does not contain K_1,3 as an induced subgraph. It is known [M.A. Henning, Graphs with large total domination number, J. Graph Theory 35(1) (2000) 21-45] that if G is a connected graph of order n with minimum degree at least two and G∉{C₃,C₅, C₆, C₁₀}, then γ_t(G)?4n/7. In this paper, we show that this upper bound can be improved if G is restricted to be a claw-free graph. We show that every connected claw-free graph G of order n and minimum degree at least two satisfies γ_t(G)?(n+2)/2 and we characterize those graphs for which γ_t(G)=⌊(n+2)/2⌋.     

Menger's Theorem for Graphs Containing no Infinite Paths

Menger's theorem can be stated as follows: Let G = (V, E) be a finite graph, and let A and B be subsets of V. Then there exists a family F of vertex-disjoint paths from A to B and a subset S of V which separates A and B, such that S consists of a choice of precisely one vertex from each path in F.Erdös conjectured that in this form the theorem can be extended to infinite graphs. We prove this to be true for graphs containing no infinite paths, by showing that in this case the problem can be reduced to the case of bipartite graphs.     

Graphs with small boundary

For a pair of vertices x and y in a graph G, we denote by d_G(x,y) the distance between x and y in G. We call x a boundary vertex of y if x and y belong to the same component and d_G(y,v)?d_G(y,x) for each neighbor v of x in G. A boundary vertex of some vertex is simply called a boundary vertex, and the set of boundary vertices in G is called the boundary of G, and is denoted by B(G).In this paper, we investigate graphs with a small boundary. Since a pair of farthest vertices are boundary vertices, |B(G)|?2 for every connected graph G of order at least two. We characterize the graphs with boundary of order at most three. We cannot give a characterization of graphs with exactly four boundary vertices, but we prove that such graphs have minimum degree at most six. Finally, we give an upper bound to the minimum degree of a connected graph G in terms of |B(G)|.     

On Toughness and Hamiltonicity of 2K2‐Free Graphs

The toughness of a (noncomplete) graph G is the minimum value of t for which there is a vertex cut A whose removal yields components. Determining toughness is an NP‐hard problem for general input graphs. The toughness conjecture of Chvátal, which states that there exists a constant t such that every graph on at least three vertices with toughness at least t is hamiltonian, is still open for general graphs. We extend some known toughness results for split graphs to the more general class of 2K₂‐free graphs, that is, graphs that do not contain two vertex‐disjoint edges as an induced subgraph. We prove that the problem of determining toughness is polynomially solvable and that Chvátal's toughness conjecture is true for 2K₂‐free graphs.