Forbidden Subgraphs Research Articles

Forbidden subgraphs of coloring graphs

Given a graph G, its k-coloring graph has vertex set given by the proper k-colorings of the vertices of G with two k-colorings adjacent if and only if they differ at exactly one vertex. Beier et al. (Discrete Math. 339:8 (2016), 2100–2112) give various characterizations of coloring graphs, including finding graphs which never arise as induced subgraphs of coloring graphs. These are called forbidden subgraphs, and if no proper subgraph of a forbidden subgraph is forbidden, it is called minimal forbidden. In this paper, we construct a finite collection of minimal forbidden subgraphs that come from modifying theta graphs. We also construct an infinite family of minimal forbidden subgraphs similar to the infinite family found by Beier et al.

Interval k-Graphs and Orders

An interval k-graph is the intersection graph of a family of intervals of the real line partitioned into k classes with vertices adjacent if and only if their corresponding intervals intersect and belong to different classes. In this paper we study the cocomparability interval k-graphs; that is, the interval k-graphs whose complements have a transitive orientation and are therefore the incomparability graphs of strict partial orders. For brevity we call these orders interval k-orders. We characterize the kind of interval representations a cocomparability interval k-graph must have, and identify the structure that guarantees an order is an interval k-order. The case k = 2 is peculiar: cocomparability interval 2-graphs (equivalently proper- or unit-interval bigraphs, bipartite permutation graphs, and complements of proper circular-arc graphs to name a few) have been characterized in many ways, but we show that analogous characterizations do not hold if k > 2. We characterize the cocomparability interval 3-graphs via one forbidden subgraph and hence interval 3-orders via one forbidden suborder.

Induced Turán Numbers

The classical Kővári–Sós–Turán theorem states that ifGis ann-vertex graph with no copy ofKs,tas a subgraph, then the number of edges inGis at mostO(n2−1/s). We prove that if one forbidsKs,tas aninducedsubgraph, and also forbidsanyfixed graphHas a (not necessarily induced) subgraph, the same asymptotic upper bound still holds, with different constant factors. This introduces a non-trivial angle from which to generalize Turán theory to induced forbidden subgraphs, which this paper explores. Along the way, we derive a non-trivial upper bound on the number of cliques of fixed order in aKr-free graph with no induced copy ofKs,t. This result is an induced analogue of a recent theorem of Alon and Shikhelman and is of independent interest.

On polygon numbers of circle graphs and distance hereditary graphs

Circle graphs are intersection graphs of chords in a circle and k-polygon graphs are intersection graphs of chords in a convex k-sided polygon where each chord has its endpoints on distinct sides. The k-polygon graphs, for k≥2, form an infinite chain of graph classes, each of which contains the class of permutation graphs. The union of all of those graph classes is the class of circle graphs. The polygon number ψ(G) of a circle graph G is the minimum k such that G is a k-polygon graph. Given a circle graph G and an integer k, determining whether ψ(G)≤k is NP-complete, while the problem is solvable in polynomial time for fixed k.In this paper, we show that ψ(G) is always at least as large as the asteroidal number of G, and equal to the asteroidal number of G when G is a connected distance hereditary graph that is not a clique. This implies that the classes of distance hereditary permutation graphs and distance hereditary AT-free graphs are the same, and we give a forbidden subgraph characterization of that class. We also establish the following upper bounds: ψ(G) is at most the clique cover number of G if G is not a clique, at most 1 plus the independence number of G, and at most ⌈n∕2⌉ where n≥3 is the number of vertices of G. Our results lead to linear time algorithms for finding the minimum number of corners that must be added to a given circle representation to produce a polygon representation, and for finding the asteroidal number of a distance hereditary graph, both of which are improvements over previous algorithms for those problems.

Forbidden Subgraph Characterizations of Extensions of Gallai Graph Operator to Signed Graph

Forbidden Subgraph Characterizations of Extensions of Gallai Graph Operator to Signed Graph

Compressed cliques graphs, clique coverings and positive zero forcing

Zero forcing parameters, associated with graphs, have been studied for over a decade, and have gained popularity as the number of related applications grows. In particular, it is well-known that such parameters are related to certain vertex coverings. Continuing along these lines, we investigate positive zero forcing within the context of certain clique coverings. A key object considered here is the compressed cliques graph. We study a number of properties associated with the compressed cliques graph, including: uniqueness, forbidden subgraphs, connections to Johnson graphs, and positive zero forcing.

Forbidden subgraphs for chorded pancyclicity

We call a graph Gpancyclic if it contains at least one cycle of every possible length m, for 3≤m≤|V(G)|. In this paper, we define a new property called chorded pancyclicity. We explore forbidden subgraphs in claw-free graphs sufficient to imply that the graph contains at least one chorded cycle of every possible length 4,5,…,|V(G)|. In particular, certain paths and triangles with pendant paths are forbidden.

Topics on data transmission problem in software definition network

Abstract In normal computer networks, the data transmission between two sites go through the shortest path between two corresponding vertices. However, in the setting of software definition network (SDN), it should monitor the network traffic flow in each site and channel timely, and the data transmission path between two sites in SDN should consider the congestion in current networks. Hence, the difference of available data transmission theory between normal computer network and software definition network is that we should consider the prohibit graph structures in SDN, and these forbidden subgraphs represent the sites and channels in which data can’t be passed by the serious congestion. Inspired by theoretical analysis of an available data transmission in SDN, we consider some computational problems from the perspective of the graph theory. Several results determined in the paper imply the sufficient conditions of data transmission in SDN in the various graph settings.

Forbidden pairs of disconnected graphs for traceability in connected graphs

Let H be a class of given graphs. A graph G is said to be H-free if G contains no induced copies of H for every H?H. A graph is called traceable if it contains a Hamilton path. Faudree and Gould (1997) characterized all the pairs {R,S} of connected subgraphs such that every connected {R,S}-free graph is traceable. Li and Vrána (2016) first consider the disconnected forbidden subgraphs, and characterized all pairs of disconnected forbidden subgraphs for hamiltonicity. In this article, we characterize all pairs {R,S} of graphs such that there exists an integer n0 such that every connected {R,S}-free graph of order at least n0 is traceable.

Minkowski complexes and convex threshold dimension

For a collection of convex bodies P1,…,Pn⊂Rd containing the origin, a Minkowski complex is given by those subsets whose Minkowski sum does not contain a fixed basepoint. Every simplicial complex can be realized as a Minkowski complex and for convex bodies on the real line, this recovers the class of threshold complexes. The purpose of this note is the study of the convex threshold dimension of a complex, that is, the smallest dimension in which it can be realized as a Minkowski complex. In particular, we show that the convex threshold dimension can be arbitrarily large. This is related to work of Chvátal and Hammer (1977) regarding forbidden subgraphs of threshold graphs. We also show that convexity is crucial this context.

Some new upper bounds of [formula omitted

The extremal number ex(n;{C3,C4}) or simply ex(n;4) denotes the maximal number of edges in a graph on n vertices with forbidden subgraphs C3 and C4. The exact number of ex(n;4) is only known for n up to 32 and n=50. There are upper and lower bounds of ex(n;4) for other values of n. In this paper, we improve the upper bound of ex(n;4) for n=33,34,…,42 and also n=d2+1 for any positive integer d≠7,57.

Hitting forbidden subgraphs in graphs of bounded treewidth

We study the complexity of a generic hitting problem H-Subgraph Hitting, where given a fixed pattern graph H and an input graph G, the task is to find a set X⊆V(G) of minimum size that hits all subgraphs of G isomorphic to H. In the colorful variant of the problem, each vertex of G is precolored with some color from V(H) and we require to hit only H-subgraphs with matching colors. Standard techniques shows that for every fixed H, the problem is fixed-parameter tractable parameterized by the treewidth of G; however, it is not clear how exactly the running time should depend on treewidth. For the colorful variant, we demonstrate matching upper and lower bounds showing that the dependence of the running time on treewidth of G is tightly governed by μ(H), the maximum size of a minimal vertex separator in H. That is, we show for every fixed H that, on a graph of treewidth t, the colorful problem can be solved in time 2O(tμ(H))⋅|V(G)|, but cannot be solved in time 2o(tμ(H))⋅|V(G)|O(1), assuming the Exponential Time Hypothesis (ETH). Furthermore, we give some preliminary results showing that, in the absence of colors, the parameterized complexity landscape of H-Subgraph Hitting is much richer.

Deleting Edges to Restrict the Size of an Epidemic: A New Application for Treewidth

Motivated by applications in network epidemiology, we consider the problem of determining whether it is possible to delete at most k edges from a given input graph (of small treewidth) so that the resulting graph avoids a set $$\mathcal {F}$$ of forbidden subgraphs; of particular interest is the problem of determining whether it is possible to delete at most k edges so that the resulting graph has no connected component of more than h vertices, as this bounds the worst-case size of an epidemic. While even this special case of the problem is NP-complete in general (even when $$h=3$$ ), we provide evidence that many of the real-world networks of interest are likely to have small treewidth, and we describe an algorithm which solves the general problem in time $$2^{O(|\mathcal {F}|w^r)}n$$ on an input graph having n vertices and whose treewidth is bounded by a fixed constant w, if each of the subgraphs we wish to avoid has at most r vertices. For the special case in which we wish only to ensure that no component has more than h vertices, we improve on this to give an algorithm running in time $$O((wh)^{2w}n)$$ , which we have implemented and tested on real datasets based on cattle movements.

Forbidden Pairs with a Common Graph Generating Almost the Same Sets

Let $\mathcal{H}$ be a family of connected graphs. A graph $G$ is said to be $\mathcal{H}$-free if $G$ does not contain any members of $\mathcal{H}$ as an induced subgraph. Let $\mathcal{F}(\mathcal{H})$ be the family of connected $\mathcal{H}$-free graphs. In this context, the members of $\mathcal{H}$ are called forbidden subgraphs.In this paper, we focus on two pairs of forbidden subgraphs containing a common graph, and compare the classes of graphs satisfying each of the two forbidden subgraph conditions. Our main result is the following: Let $H_{1},H_{2},H_{3}$ be connected graphs of order at least three, and suppose that $H_{1}$ is twin-less. If the symmetric difference of $\mathcal{F}(\{H_{1},H_{2}\})$ and $\mathcal{F}(\{H_{1},H_{3}\})$ is finite and the tuple $(H_{1};H_{2},H_{3})$ is non-trivial in a sense, then $H_{2}$ and $H_{3}$ are obtained from the same vertex-transitive graph by successively replacing a vertex with a clique and joining the neighbors of the original vertex and the clique. Furthermore, we refine a result in [Combin. Probab. Comput. 22 (2013) 733–748] concerning forbidden pairs.

Characterization of forbidden subgraphs for the existence of even factors in a graph

In 2008, it is shown that if H≅K1,3 (the claw), then every H-free graph G has an even factor if and only if δ(G)≥2 and every odd branch-bond of G having an edge-branch. In this paper, we characterize all connected graphs H with order at least three such that every H-free graph has an even factor if and only if δ(G)≥2 and every odd branch-bond of G has an edge-branch.

On a relation between [formula omitted]-path partition and [formula omitted]-path vertex cover

The minimum vertex cover problem and the minimum vertex partition problem are central problems in graph theory and many generalisations are known. Two examples are the minimumk-path vertex cover problem (MkPVCP for short), which asks for a minimum set of vertices covering every path of order k, and the minimumk-path partition problem (MkPPP for short), which asks for a minimum set of paths in a maximal path packing whose every path has at most k vertices.In this paper, we will present a relation between the MkPPP and the MkPVCP. Based on it, we will obtain new bounds for their invariants and a new sufficient condition for NP-hardness of the MkPVCP in terms of forbidden subgraphs.

Forbidden pairs for spanning (closed) trails

In Faudree and Gould (1997), the authors determined all pairs of connected graphs {H1,H2} such that any connected {H1,H2}-free graph has a spanning path (cycle), i.e., hamiltonian path (cycle). In this paper, we consider a similar problem and determine all pairs of forbidden subgraphs guaranteeing the existence of spanning (closed) trails of connected graphs. Our results show that although the forbidden pairs for the existence of spanning trails are the same as the existence of spanning paths, the forbidden pairs for the existence of spanning closed trails (supereulerian) are much different from those for the existence of spanning cycles (hamiltonian).

Lower bounds on the leaf number in graphs with forbidden subgraphs

Let G be a simple, connected graph. The leaf number, L(G) of G, is defined as the maximum number of leaf vertices contained in a spanning tree of G. Assume that G is a triangle−free graph with minimum degree δ, order n and leaf number L(G). We show that for δ = 4 and δ = 5, where cδ is a constant that depends on δ only. Similar bounds are shown to hold for triangle−free and C4−free graphs.

Complexity classification of the edge coloring problem for a family of graph classes

AbstractA class of graphs is called monotone if it is closed under deletion of vertices and edges. Any such class may be defined in terms of forbidden subgraphs. The chromatic index of a graph is the smallest number of colors required for its edge-coloring such that any two adjacent edges have different colors. We obtain a complete classification of the complexity of the chromatic index problem for all monotone classes defined in terms of forbidden subgraphs having at most 6 edges or at most 7 vertices.

Regular graphs with forbidden subgraphs of K<SUB align="right">n with k edges

In this paper, we raise a variant of a classic problem in an extremal graph theory, which is motivated by a design of fractional repetition codes, a model in distributed storage systems. For any feasible positive integers d > 2, n > 2 and k, what is the minimum possible number of vertices in a d-regular undirected graph whose subgraphs with n vertices contain at most k edges? The goal of this paper is to give the exact number of vertices for each instance of the problem and to provide some bounds for general values of n, d and k. A few general bounds with some exact values, for this Turan-type problem, are given. We present an almost complete solution for 2 < n < 6.