Set Of Finite Graphs Research Articles

Automated testing and interactive construction of unavoidable sets for graph classes of small path‐width

AbstractLet be a class of graphs with a membership test, , and let be the class of graphs in of path‐width at most . We present an interactive framework that finds an unavoidable set for , which is a set of graphs such that any graph in contains an isomorphic copy of a graph in . At the core of our framework is an algorithm that verifies whether a set of graphs is, indeed, unavoidable for . While obstruction sets are well‐studied, so far there is no general theory or algorithm for finding unavoidable sets. In general, it is undecidable whether a finite set of graphs is unavoidable for a given graph class. However, we give a criterion for termination: our algorithm terminates whenever is locally checkable of bounded maximum degree and is a finite set of connected graphs. For example, ‐regular graphs, ‐colourable graphs, and ‐free graphs are locally checkable classes. We put special emphasis on the case that is the class of cubic graphs and tailor the algorithm to this case. In particular, we introduce the new concept of high‐degree‐first path‐decompositions, which enables highly efficient pruning techniques. We exploit our framework to prove a new lower bound on the path‐width of cubic graphs. Moreover, we determine the extremal girth values of cubic graphs of path‐width for all and all smallest graphs which take on these extremal girth values. Further, we present a new constructive characterisation of the extremal cubic graphs of path‐width 3 and girth 4.

Zero-divisor graph of semisimple group-rings

Let [Formula: see text], [Formula: see text], [Formula: see text] denote the zero-divisor graph, compressed zero-divisor graph and annihilating ideal graph of a commutative ring [Formula: see text], respectively. In this paper, we prove that [Formula: see text] for a semisimple commutative ring [Formula: see text] and represent [Formula: see text] as a generalized join of a finite set of graphs. Further, we study the zero-divisor graph of a semisimple group-ring [Formula: see text] and proved several structural properties of [Formula: see text] and [Formula: see text], where [Formula: see text] is a field with [Formula: see text] elements and [Formula: see text] is a cyclic group with [Formula: see text] elements.

Estimating parameters associated with monotone properties

AbstractThere has been substantial interest in estimating the value of a graph parameter, i.e. of a real-valued function defined on the set of finite graphs, by querying a randomly sampled substructure whose size is independent of the size of the input. Graph parameters that may be successfully estimated in this way are said to be testable or estimable, and the sample complexity qz = qz(ε) of an estimable parameter z is the size of a random sample of a graph G required to ensure that the value of z(G) may be estimated within an error of ε with probability at least 2/3. In this paper, for any fixed monotone graph property $\mathcal{P}= \text{Forb}\!(\mathcal{F}),$ we study the sample complexity of estimating a bounded graph parameter z that, for an input graph G, counts the number of spanning subgraphs of G that satisfy$\mathcal{P}$. To improve upon previous upper bounds on the sample complexity, we show that the vertex set of any graph that satisfies a monotone property $\mathcal{P}$ may be partitioned equitably into a constant number of classes in such a way that the cluster graph induced by the partition is not far from satisfying a natural weighted graph generalization of $\mathcal{P}$. Properties for which this holds are said to be recoverable, and the study of recoverable properties may be of independent interest.

Sparse obstructions for minor-covering parameters

Given a finite set of graphs H and a non-negative integer k, we define Ak(H) as the set containing every graph G that has k vertices whose removal provides a graph without any of the graphs in H as a minor. It is known that if H contains at least one planar graph then each obstruction in Ak(H) has at most kcH vertices, for some cH depending only on the choice of H. In this paper, we investigate the size of the graphs in Ak(H) that belong to certain classes of sparse graphs. In particular, we prove that for every graph F, if H contains at least one planar graph and only connected graphs, all graphs in Ak(H) that are F-topological minor-free have at most cF,H⋅k vertices, where cF,H depends exclusively on the choice of H and F. Our result is a consequence of two more general conditions on graph parameters, namely the Finite Integer Index Property and Protrusion Decomposability, that can serve as a general framework for proving linear bounds for obstructions.

Measures on the square as sparse graph limits

We study a metric on the set of finite graphs in which two graphs are considered to be similar if they have similar bounded dimensional “factors”. We show that limits of convergent graph sequences in this metric can be represented by symmetric Borel measures on [0,1]2. This leads to a generalization of dense graph limit theory to sparse graph sequences.

On zero-free intervals of flow polynomials

For any integer k≥0, let ξk be the supremum in (1,2] such that the flow polynomial F(G,λ) has no real roots in (1,ξk) for all graphs G with at most k vertices of degrees larger than 3. We prove that ξk can be determined by considering a finite set of graphs and show that ξk=2 for k≤2, ξ3=1.430⋯ , ξ4=1.361⋯ and ξ5=1.317⋯ .

Effective computation of immersion obstructions for unions of graph classes

In the final paper of the Graph Minors series Robertson and Seymour proved that graphs are well-quasi-ordered under the immersion ordering. A direct implication of this theorem is that each class of graphs that is closed under taking immersions can be fully characterized by forbidding a finite set of graphs (immersion obstruction set). However, as the proof of the well-quasi-ordering theorem is non-constructive, there is no generic procedure for computing such a set. Moreover, it remains an open issue to identify for which immersion-closed graph classes the computation of those sets can become effective. By adapting the tools that were introduced by Adler, Grohe and Kreutzer, for the effective computation of minor obstruction sets, we expand the horizon of computability to immersion obstruction sets. In particular, our results propagate the computability of immersion obstruction sets of immersion-closed graph classes to immersion obstruction sets of finite unions of immersion-closed graph classes.

Finite dualities and map-critical graphs on a fixed surface

Let K be a class of graphs. A pair ( F , U ) is a finite duality in K if U ∈ K , F is a finite set of graphs, and for any graph G in K we have G ⩽ U if and only if F ⩽̸ G for all F ∈ F , where “⩽” is the homomorphism order. We also say U is a dual graph in K . We prove that the class of planar graphs has no finite dualities except for two trivial cases. We also prove that the class of toroidal graphs has no 5-colorable dual graphs except for two trivial cases. In a sharp contrast, for a higher genus orientable surface S we show that Thomassenʼs result (Thomassen, 1997 [17]) implies that the class, G ( S ) , of all graphs embeddable in S has a number of finite dualities. Equivalently, our first result shows that for every planar core graph H except K 1 and K 4 , there are infinitely many minimal planar obstructions for H-coloring (Hell and Nešetřil, 1990 [4]), whereas our later result gives a converse of Thomassenʼs theorem (Thomassen, 1997 [17]) for 5-colorable graphs on the torus.

On graph equivalences preserved under extensions

Let G be the set of finite graphs whose vertices belong to some fixed countable set, and let ≡ be an equivalence relation on G . By the strengthening of ≡ we mean an equivalence relation ≡ s such that G ≡ s H , where G , H ∈ G , if for every F ∈ G , G ∪ F ≡ H ∪ F . The most important case that we study in this paper concerns equivalence relations defined by graph properties. We write G ≡ Φ H , where Φ is a graph property and G , H ∈ G , if either both G and H have the property Φ , or both do not have it. We characterize the strengthening of the relations ≡ Φ for several graph properties Φ . For example, if Φ is the property of being a k -connected graph, we find a polynomially verifiable (for k fixed) condition that characterizes the pairs of graphs equivalent with respect to ≡ s Φ . We obtain similar results when Φ is the property of being k -colorable, edge 2 -colorable, Hamiltonian, or planar, and when Φ is the property of containing a subgraph isomorphic to a fixed graph H . We also prove several general theorems that provide conditions for ≡ s to be of some specific form. For example, we find a necessary and sufficient condition for the relation ≡ s to be the identity. Finally, we make a few observations on the strengthening in a more general case when G is the set of finite subsets of some countable set.

Learning block-preserving graph patterns and its application to data mining

Recently, due to the rapid growth of electronic data having graph structures such as HTML and XML texts and chemical compounds, many researchers have been interested in data mining and machine learning techniques for finding useful patterns from graph-structured data (graph data). Since graph data contain a huge number of substructures and it tends to be computationally expensive to decide whether or not such data have given structural features, graph mining problems face computational difficulties. Let ${\mathcal{C}}$ be a graph class which satisfies a connected hereditary property and contains infinitely many different biconnected graphs, and for which a special kind of the graph isomorphism problem can be computed in polynomial time. In this paper, we consider learning and mining problems for ${\mathcal{C}}$ . Firstly, we define a new graph pattern, which is called a block preserving graph pattern (bp-graph pattern) for ${\mathcal{C}}$ . Secondly, we present a polynomial time algorithm for deciding whether or not a given bp-graph pattern matches a given graph in ${\mathcal{C}}$ . Thirdly, by giving refinement operators over bp-graph patterns, we present a polynomial time algorithm for finding a minimally generalized bp-graph pattern for ${\mathcal{C}}$ . Outerplanar graphs are planar graphs which can be embedded in the plane in such a way that all of vertices lie on the outer boundary. Many pharmacologic chemical compounds are known to be represented by outerplanar graphs. The class of connected outerplanar graphs ${\mathcal{O}}$ satisfies the above conditions for ${\mathcal{C}}$ . Next, we propose two incremental polynomial time algorithms for enumerating all frequent bp-graph patterns with respect to a given finite set of graphs in ${\mathcal{O}}$ . Finally, by reporting experimental results obtained by applying the two graph mining algorithms to a subset of the NCI dataset, we evaluate the performance of the two graph mining algorithms.

Finite duality for some minor closed classes

Abstract Let K be a class of finite graphs and F = { F 1 , F 2 , … , F m } be a set of finite graphs. Then, K is said to have finite-duality if there exists a graph U in K such that for any graph G in K , G is homomorphic to U if and only if F i is not homomorphic to G, for all i = 1 , 2 , … , m . Nesetřil asked in [J. Nesetřil, Homonolo Combinatorics Workshop, Nova Louka, Czech Rep., (2003)] if non-trivial examples can be found. In this note, we answer this positively by showing classes containing arbitrary long anti-chains and yet having the finite-duality property.

Graph decompositions for cartesian products

Abstract In this paper we describe an algorithmic construction that, given a tree-decomposition of a graph G and a path-decomposition of a graph H , provides a tree-decomposition of the cartesian product of G and H . From the latter, we derive upper bounds on the treewidth and on the pathwidth of the cartesian product, expressed in terms of the treewidth and of the pathwidth of the two involved graphs. On the other hand, in the context of graph grammars and graph logic, we prove that the cartesian product of a class of graphs by a finite set of graphs preserves the property of being a context-free set, and if the graphs in the finite set are all connected, the property of being an MS -definable set is also preserved.

The recognizability of sets of graphs is a robust property

Once the set of finite graphs is equipped with an algebra structure (arising from the definition of operations that generalize the concatenation of words), one can define the notion of a recognizable set of graphs in terms of finite congruences. Applications to the construction of efficient algorithms and to the theory of context-free sets of graphs follow naturally. The class of recognizable sets depends on the signature of graph operations. We consider three signatures related respectively to Hyperedge Replacement ( HR) context-free graph grammars, to Vertex Replacement ( VR) context-free graph grammars, and to modular decompositions of graphs. We compare the corresponding classes of recognizable sets. We show that they are robust in the sense that many variants of each signature (where in particular operations are defined by quantifier-free formulas, a quite flexible framework) yield the same notions of recognizability. We prove that for graphs without large complete bipartite subgraphs, HR-recognizability and VR-recognizability coincide. The same combinatorial condition equates HR-context-free and VR-context-free sets of graphs. Inasmuch as possible, results are formulated in the more general framework of relational structures.

Graph family operations

In previous papers, Catlin introduced four functions, denoted S O , S R , S C , and S H , between sets of finite graphs. These functions proved to be very useful in establishing properties of several classes of graphs, including supereulerian graphs and graphs with nowhere zero k-flows for a fixed integer k⩾3. Unfortunately, a subtle error caused several theorems previously published in Catlin (Discrete Math. 160 (1996) 67–80) to be incorrect. In this paper we correct those errors and further explore the relations between these functions, showing that there is a sort of duality between them and that they act as inverses of one another on certain sets of graphs.

Polygonal complexes I: combinatorial and geometric properties

Given a finite set of graphs (that can be either finite or infinite), we construct polygonal complexes such that the link of each vertex is isomorphic to one of the give graphs. For this purpose, we use, essentially, the theory of complexes of groups (see Corson, 1992; Haefliger, 1991; Stallings, 1991).

Proportional graphs

AbstractLet S be a finite set of graphs and t a real number, 0 < t < 1. A (deterministic) graph G is (t, 5)‐proportional if for every H ∈ S, the number of induced subgraphs of G isomorphic to H equals the expected number of induced copies of H in the random graph Gn, t where n = |V(G)|. Let Sk = {all graphs on k vertices}, in particular S3 = {K3, P2, K2Kt, D3}. The notion of proportional graphs stems from the study of random graphs (Barbour, Karoński, and Ruciński, J Combinat. Th. Ser. B, 47, 125‐145, 1989; Janson and Nowicki, Prob. Th. Rel. Fields, to appear, Janson, Random Struct. Alg., 1, 15‐37, 1990) where it is shown that (t, S3)‐proportional graphs play a very special role; we thus call them simply t‐proportional. However, only a few ½‐proportional graphs on 8 vertices were known and it was an open problem whether there are any f‐proportional graphs with t ≠ ½ at all. In this paper, we show that there are infinitely many ½‐proportional graphs and that there are t‐proportional graphs with t≠. Both results are proved constructively. [We are not able to provide the latter construction for all f∈ Q∩(0,1), but the set of ts for which our construction works is dense in (0,1).] To support a conviction that the existence of (t, S3)‐proportional graphs was not quite obvious, we show that there are no (t, S4)‐proportional graphs.

𝒢‐constructibility of planar graphs

AbstractIn this paper, the concept of the 𝒢‐constructibility of graphs is introduced and investigated with particular reference to planar graphs. It is conjectured that the planar graphs are minimally N‐constructible, where N is a finite set of graphs and an infinite set 𝒢 is obtained such that the planar graphs are also minimally 𝒢‐constructible. Finally, some properties of the set of all N‐constructible graphs are discussed and compared with the corresponding properties of planar graphs.

Zum Basisproblem der nicht in die projektive Ebene einbettbaren Graphen II

Let Γ be the set of finite graphs which are not embeddable into the projective plane. Let M( Γ >) denote the set of the minimal graphs of Γ. In former papers [3] and [5] K. Wagner has proved the following result: Each minimal graph G of M( Γ >) which is different from the graphs G 1, G 2, …, G 12, G 14, D, F fulfils the three following conditions: 1. (1) G is at least 3-vertex connected. 2. (2) In G exists one vertex v, for which the graph G v obtained from G by deleting v and all edges of G which are incident with v contains a U( K 3,3). 3. (3) For each vertex v of G, the graph G v is either planar or contains a U( K 3,3). In the following we investigate the character of these graphs G. For this some essential terms are necessary. Let G be a graph with the vertex set V and G′ a subgraph of G with the vertex set V′. Then G′ is called a subgraph of the nth class (designated by G′ ⊆ ( n) G ) if and only if the chromatic number of the subgraph G( V − V′) of G spanned by V − V′ is equal to n. Let X be a connected component of the spanned subgraph G( V − V′) of G and let Y be the bipartite graph consisting of all those edges of G which combine X with G′. Then the graph Q = X ⌣ Y is called a relative component of G in relation to G′ (or “Querstück” of G in relation to G′). Let z 1, z 2, z 3, z 4, a, b be six different vertices of G from which z 1, z 2, z 3, z 4 lie on a circuit Z of G. Let W 1, W 2, W 3 denote three paths of G with W 1 = W( z 1, z 2), W 2 = W( z 3, z 4), and W 3 = W( a, b) 1 1 W( z 1, z 2) means that z 1 and z 2 are the endpoints of this path. Similar: W( z 3, z 4) and W( a, b). . The subgraph W 1 ⌣ W 2 ⌣ W 3 of G is called a “Kreuzhaube” H with the basis points z 1, z 2, z 3, z 4 if and only if 1. (1) W 1 ⌣ Z = {z 1, z 2} , 2. (2) W 2 ⌣ Z = {z 3, z 4} , 3. (3) W 1 ⌣ W 2 = W 3 ⌣ Z = ⊘ , 4. (4) W 3 ⌣ W 1 = {a} , 5. (5) W 3 ⌣ W 2 = {b} , 6. (6) The two pairs of basis points z 1, z 2 and z 3, z 4 are separated from each other on Z (Fig. 1). ▪ Having characterized minimal graphs of M( Γ >) in “Satz 1”, we prove the following theorem: Let G be a minimal graph of M(Γ >) which is different from G 1,G 2,…,G 12 and G 14 and which contains a G′ = U(K 3,3) as a subgraph of the nth class with n ≥ 1. Then only one of the following three statements holds for each of the subgraphs G′ = U(K 3,3) and for each relative component Q of G in relation to G′ : 1. (1) Q is a star (“Stern”) . 2. (2) Q contains a “Kreuzhaube” H which has only its four basis-points with G′ = U(K 3,3) in common. To be more precise: If all those vertices of Q which do not lie on G′ are contracted in one single vertex, the result is that Q becomes a star Q ∗ and G becomes G ∗ (with fewer vertices than G and therefore G ∗ being embeddable into the projective plane), then Q in relation with each embedding of G ∗ into the projective plane contains a “Kreuzhaube” which has only its four basis-points in common with that circuit of the G′ on which the endpoints of Q ∗ lie at the embedding of G ∗ . 3. (3) G = P. In this equation P means the graph of M(Γ >) which one obtains from two disjoint K 3,3 by identifying two neighboring edges of the one K 3,3 with two neighboring edges of the other K 3,3.