Diamond-free Graphs Research Articles

Connected [formula omitted]-free graphs whose second largest eigenvalue does not exceed 1

Connected graphs whose second largest eigenvalue λ2 does not exceed 1 have been investigated in the last four decades. Over the years only few particular classes with this spectral property are completely determined. For example, they include bipartite graphs, line graphs and threshold graphs. In this paper we determine all connected (K4−e)-free graphs, i.e., the graphs that do not contain an induced subgraph isomorphic to the graph obtained by removing an edge from the complete graph of order 4; such graphs are also called diamond-free graphs. Since every triangle-free graph is automatically diamond-free, our results includes all triangle-free graphs with λ2≤1. Moreover, it includes all connected bipartite graphs with the same spectral property, and therefore strengthens the result of Petrović (J. Combin. Theory B, 1991).

Graphs with Two Moplexes

Moplexes are natural graph structures that arise when lifting Dirac’s classical theorem from chordal graphs to general graphs. The notion is known to be closely related to lexicographic searches in graphs as well as to asteroidal triples, and has been applied in several algorithms related to graph classes such as interval graphs, claw-free, and diamond-free graphs. However, while every non-complete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is, among others, motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are interval.In this work we initiate an investigation of k-moplex graphs, which are defined as graphs containing at most k moplexes. Of particular interest is the smallest nontrivial case, k = 2, which forms a counterpart to the class of interval graphs. As our main structural result, we show that the class of connected 2-moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely Graph Isomorphism and Max-Cut. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets. We also discuss sufficient conditions for the existence of Hamiltonian paths in 2-moplex graphs as well as connections with avoidable vertices.

Strong cliques in diamond-free graphs

A strong clique in a graph is a clique intersecting all inclusion-maximal stable sets. Strong cliques play an important role in the study of perfect graphs. We study strong cliques in the class of diamond-free graphs, from both structural and algorithmic points of view. We show that the following five NP-hard or co-NP-hard problems all remain NP-hard or co-NP-hard when restricted to the class of diamond-free graphs: Is a given clique strong? Does the graph have a strong clique? Is every vertex contained in a strong clique? Given a partition of the vertex set into cliques, is every clique in the partition strong? Can the vertex set be partitioned into strong cliques?On the positive side, we show that the following three problems whose computational complexity is open in general can be solved in polynomial time in the class of diamond-free graphs: Does every induced subgraph have a strong clique? Is every maximal clique strong? Is every edge contained in a strong clique? The last two results are derived from a characterization of diamond-free graphs in which every maximal clique is strong, which also implies an improved Erdős-Hajnal property for such graphs.

A new approximate cluster deletion algorithm for diamond-free graphs

The cluster deletion problem (CD) asks for transforming a given graph into a disjoint union of cliques by removing as few edges as possible. CD is among the most studied combinatorial optimization problem and, for general graphs, it is NP-hard. In the present paper, we identify a new polynomially solvable CD subproblem. We specifically propose a two-phase polynomial-time algorithm that optimally solves CD on the class of (butterfly,diamond)-free graphs. For this latter class of graphs, our two-phase algorithm provides optimal solutions even for another clustering variant, namely, cluster editing. Then, we propose a 2-optimal CD algorithm dedicated to the super-class of diamond-free graphs. For this class, we also show that CD, when parameterised by the number of deleted edges, admits a quadratic-size kernel. Finally, we report the results of experiments carried out on numerous diamond-free graphs, showing the effectiveness of the proposed approximate algorithm in terms of solution quality.

On Total Domination and Hop Domination in Diamond-Free Graphs

Two vertices in a graph are said to 2-step dominate each other if they are at distance 2 apart. A set S of vertices in a graph G is a hop dominating set of G if every vertex outside S is 2-step dominated by some vertex of S. The hop domination number, $$\gamma _{h}(G)$$, of G is the minimum cardinality of a hop dominating set of G. A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to at least one vertex of S. The total domination number, $$\gamma _{t}(G)$$, of G is the minimum cardinality of a total dominating set of G. It is known that if G is a triangle-free graph, then $$\gamma _{h}(G)\le \gamma _{t}(G)$$. But there are connected graphs G for which the difference $$\gamma _{h}(G)-\gamma _{t}(G)$$ can be made arbitrarily large. It would be interesting to find other classes of graphs G that satisfy $$\gamma _{h}(G)\le \gamma _{t}(G)$$. In this paper, we study the relationship between total domination number and hop domination number in diamond-free graph. We prove that if G is diamond-free graph of order n with the exception of two special graphs, then $$\gamma _{h}(G)- \gamma _{t}(G)\le \frac{n}{6}$$. Furthermore, we find two subclasses of diamond-free graphs G that satisfy $$\gamma _{h}(G)\le \gamma _{t}(G)$$ and generalize the known result.

Approximating weighted neighborhood independent sets

A neighborhood independent set (NI-set) is a subset of edges in a graph such that the closed neighborhood of any vertex contains at most one edge of the subset. Finding a maximum cardinality NI-set is an NP-complete problem. We consider the weighted version of this problem. For general graphs we give an algorithm with approximation ratio Δ, and for diamond-free graphs we give a ratio Δ/2+1, where Δ is the maximum degree of the input graph. Furthermore, we show that the problem is polynomially solvable on cographs. Finally, we give a tight upper bound on the cardinality of a NI-set on regular graphs.

Dealing with several parameterized problems by random methods

In this paper, we apply random methods to deal with several parameterized problems. For the Parameterized Weighted P3-Packing problem, by randomly partitioning the vertices in given graph, a tripartite graph can be obtained. We prove that the Parameterized Weighted P3-Packing problem can be solved in polynomial time on tripartite graphs. Based on the algorithm on tripartite graphs, a randomized parameterized algorithm of running time O⁎(32k) is given for the Parameterized Weighted P3-Packing problem. For the Parameterized Weighted Load Coloring problem, by randomly partitioning the vertices in given graph into two parts and studying the structure properties of the connected components in two parts, a randomized parameterized algorithm of running time O⁎(11.32k) is presented. For the Parameterized Claw-free Edge Deletion problem on Diamond-free Graphs, by combining random with branching methods, a parameterized algorithm of running time O⁎(2.42k) is given.

Colouring diamond-free graphs

The Colouring problem is that of deciding, given a graph G and an integer k, whether G admits a (proper) k-colouring. For all graphs H up to five vertices, we classify the computational complexity of Colouring for (diamond,H)-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for (diamond,P1+2P2)-free graphs. Our technique for handling this case is to reduce the graph under consideration to a k-partite graph that has a very specific decomposition. As a by-product of this general technique we are also able to prove boundedness of clique-width for four other new classes of (H1,H2)-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of (H1,H2)-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8.

A Polynomial Kernel for Block Graph Deletion

In the Block Graph Deletion problem, we are given a graph G on n vertices and a positive integer k, and the objective is to check whether it is possible to delete at most k vertices from G to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with $${\mathcal {O}}(k^{6})$$ vertices for the Block Graph Deletion problem. This is a first step to investigate polynomial kernels for deletion problems into non-trivial classes of graphs of bounded rank-width, but unbounded tree-width. Our result also implies that Chordal Vertex Deletion admits a polynomial-size kernel on diamond-free graphs. For the kernelization and its analysis, we introduce the notion of ‘complete degree’ of a vertex. We believe that the underlying idea can be potentially applied to other problems. We also prove that the Block Graph Deletion problem can be solved in time $$10^{k}\cdot n^{{\mathcal {O}}(1)}$$ .

On probe 2-clique graphs and probe diamond-free graphs

Graph Theory Given a class G of graphs, probe G graphs are defined as follows. A graph G is probe G if there exists a partition of its vertices into a set of probe vertices and a stable set of nonprobe vertices in such a way that non-edges of G, whose endpoints are nonprobe vertices, can be added so that the resulting graph belongs to G. We investigate probe 2-clique graphs and probe diamond-free graphs. For probe 2-clique graphs, we present a polynomial-time recognition algorithm. Probe diamond-free graphs are characterized by minimal forbidden induced subgraphs. As a by-product, it is proved that the class of probe block graphs is the intersection between the classes of chordal graphs and probe diamond-free graphs.

Minimally Unbalanced Diamond-Free Graphs and Dyck-Paths

A $\{0,1\}$-matrix $\mathsf{A}$ is balanced if it does not contain a submatrix of odd order having exactly two 1's per row and per column. A graph is balanced if its clique-matrix is balanced. No characterization of minimally unbalanced graphs is known, and even no conjecture on the structure of such graphs has been posed, contrary to what happened for perfect graphs. In this paper, we provide such a characterization for the class of diamond-free graphs and establish a connection between minimally unbalanced diamond-free graphs and Dyck-paths.

Clique-perfectness and balancedness of some graph classes

A graph is clique-perfect if the maximum size of a clique-independent set (a set of pairwise disjoint maximal cliques) and the minimum size of a clique-transversal set (a set of vertices meeting every maximal clique) coincide for each induced subgraph. A graph is balanced if its clique-matrix contains no square submatrix of odd size with exactly two ones per row and column. In this work, we give linear-time recognition algorithms and minimal forbidden induced subgraph characterizations of clique-perfectness and balancedness of P4-tidy graphs and a linear-time algorithm for computing a maximum clique-independent set and a minimum clique-transversal set for any P4-tidy graph. We also give a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for balancedness of paw-free graphs. Finally, we show that clique-perfectness of diamond-free graphs can be decided in polynomial time by showing that a diamond-free graph is clique-perfect if and only if it is balanced.

Mixed unit interval graphs

The class of intersection graphs of unit intervals of the real line whose ends may be open or closed is a strict superclass of the well-known class of unit interval graphs. We pose a conjecture concerning characterizations of such mixed unit interval graphs, verify parts of it in general, and prove it completely for diamond-free graphs. In particular, we characterize diamond-free mixed unit interval graphs by means of an infinite family of forbidden induced subgraphs, and we show that a diamond-free graph is mixed unit interval if and only if it has intersection representations using unit intervals such that all ends of the intervals are integral.

Arboricity, [formula omitted]-index, and dynamic algorithms

We propose a new data structure for manipulating graphs, called h-graph, which is particularly suited for designing dynamic algorithms. The structure itself is simple, consisting basically of a triple of elements, for each vertex of the graph. The overall size of all triples is O(n+m), for a graph with n vertices and m edges. We describe algorithms for performing the basic operations related to dynamic applications, as insertions and deletions of vertices or edges, and adjacency queries. The data structure employs a technique first described by Chiba and Nishizeki [Chiba, Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput. 14 (1) (1985) 210–223], and relies on the arboricity of graphs. Using the proposed data structure, we describe several dynamic algorithms for solving problems as listing the cliques of a given size, recognizing diamond-free graphs, and finding simple, simplicial and dominated vertices. These algorithms are the first of their kind to be proposed in the literature. In fact, the dynamic algorithms for the above problems lead directly to new static algorithms, and using the data structure we also design new static algorithms for the problems of counting subgraphs of size 4, recognizing cop-win graphs and recognizing strongly chordal graphs. The complexities of all of the proposed static algorithms improve over the complexities of the so far existing algorithms, for graphs of low arboricity. In addition, for the problems of counting subgraphs of size 4 and recognizing diamond-free graphs, the improvement is general.

Clique separator decomposition of hole-free and diamond-free graphs and algorithmic consequences

Clique separator decomposition, introduced by Whitesides and Tarjan, is one of the most important graph decompositions. A hole is a chordless cycle with at least five vertices. A paraglider is a graph with five vertices a,b,c,d,e and edges ab,ac,bc,bd,cd,ae,de. We show that every (hole, paraglider)-free graph admits a clique separator decomposition into graphs of three very specific types. This yields efficient algorithms for various optimization problems in this class of graphs.

Some properties of edge intersection graphs of single-bend paths on a grid

In this paper we consider graphs G whose vertices can be represented as single-bend paths (i.e., paths with at most one turn) on a rectangular grid, such that two vertices are adjacent in G if and only if the corresponding paths share at least one edge of the grid. These graphs, called B 1 -EPG graphs, were first introduced in Golumbic et al. (2009) [13]. Here we show that the neighborhood of every vertex in a B 1 -EPG graph induces a weakly chordal graph. From this we conclude that the family F of B 1 -EPG graphs satisfies the Erdős–Hajnal property with ϵ ( F ) = 1 3 , i.e., that every B 1 -EPG graph on n vertices contains either a clique or a stable set of size at least n 1 3 . Finally we give a characterization of B 1 -EPG graphs among some subclasses of chordal graphs, namely chordal bull-free graphs, chordal claw-free graphs, chordal diamond-free graphs, and special cases of split graphs.

Unique intersectability of diamond-free graphs

For a graph G with vertices v 1 , v 2 , … , v n , a simple set representation of G is a family F = { S 1 , S 2 , … , S n } of distinct nonempty sets such that | S i ∩ S j | = 1 if v i v j is an edge in G , and | S i ∩ S j | = 0 otherwise. Let S ( F ) = ⋃ i = 1 n S i , and let ω s ( G ) denote the minimum | S ( F ) | of a simple set representation F of G . If, for every two minimum simple set representations F and F ′ of G , F can be obtained from F ′ by a bijective mapping from S ( F ′ ) to S ( F ) , then G is said to be s -uniquely intersectable. In this paper, we are concerned with the s -unique intersectability of diamond-free graphs, where a diamond is a K 4 with one edge deleted. Moreover, for a diamond-free graph G , we also derive a formula for computing ω s ( G ) .

Formula omitted]-partition of some classes of graphs

We consider the problem of partitioning the vertex-set of a graph into four non-empty sets A,B,C,D such that every vertex of A is adjacent to every vertex of B and every vertex of C is adjacent to every vertex of D. The complexity of deciding if a graph admits such a partition is unknown. We show that it can be solved in polynomial time for several classes of graphs: K4-free graphs, diamond-free graphs, planar graphs, graphs with bounded treewidth, claw-free graphs, (C5,P5)-free graphs and graphs with few P4’s.

On the connectivity of [formula omitted]-diamond-free graphs

For a vertex v of a graph G , we denote by d ( v ) the degree of v . The local connectivity κ ( u , v ) of two vertices u and v in a graph G is the maximum number of internally disjoint u – v paths in G , and the connectivity of G is defined as κ ( G ) = min { κ ( u , v ) | u , v ∈ V ( G ) } . Clearly, κ ( u , v ) ≤ min { d ( u ) , d ( v ) } for all pairs u and v of vertices in G . Let δ ( G ) be the minimum degree of G . We call a graph G maximally connected when κ ( G ) = δ ( G ) and maximally local connected when κ ( u , v ) = min { d ( u ) , d ( v ) } for all pairs u and v of distinct vertices in G . For an integer p ≥ 2 , we define a p - diamond as the graph with p + 2 vertices, where two adjacent vertices have exactly p common neighbors, and the graph contains no further edges. For p = 2 , the 2-diamond is the usual diamond. We call a graph p - diamond-free if it contains no p -diamond as a (not necessarily induced) subgraph. In 2007, Dankelmann, Hellwig and Volkmann [P. Dankelmann, A. Hellwig, L. Volkmann, On the connectivity of diamond-free graphs, Discrete Appl. Math. 155 (2007), 2111–2117] proved that a connected diamond-free graph G of order n ≤ 3 δ ( G ) is maximally connected when δ ( G ) ≥ 3 . In this paper, we show that such graphs are even maximally local connected when n ≤ 3 δ ( G ) − 1 . Examples demonstrate that this bound is best possible. In addition, we present similar results for p -diamond-free graphs.

On minimally b-imperfect graphs

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbour in all other color classes. The b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph H of G . A graph is minimally b-imperfect if it is not b-perfect and every proper induced subgraph is b-perfect. We give a list F of minimally b-imperfect graphs, conjecture that a graph is b-perfect if and only if it does not contain a graph from this list as an induced subgraph, and prove this conjecture for diamond-free graphs, and graphs with chromatic number at most three.