Isometric Subgraphs Research Articles

Optimization in graphical small cancellation theory

Gromov (2003) [5] constructed finitely generated groups whose Cayley graphs contain all graphs from a given infinite sequence of expander graphs of unbounded girth and bounded diameter-to-girth ratio. These so-called Gromov monster groups provide examples of finitely generated groups that do not coarsely embed into Hilbert space, among other interesting properties. If graphs in Gromov's construction admit graphical small cancellation labellings, then one gets similar examples of Cayley graphs containing all the graphs of the family as isometric subgraphs. Osajda (2020) [11] recently showed how to obtain such labellings using the probabilistic method. In this short note, we simplify Osajda's approach, decreasing the number of generators of the resulting group significantly.

Factorization and pseudofactorization of weighted graphs.

For unweighted graphs, finding isometric embeddings of a graph is closely related to decompositions of into Cartesian products of smaller graphs. When is isomorphic to a Cartesian graph product, we call the factors of this product a factorization of . When is isomorphic to an isometric subgraph of a Cartesian graph product, we call those factors a pseudofactorization of . Prior work has shown that an unweighted graph's pseudofactorization can be used to generate a canonical isometric embedding into a product of the smallest possible pseudofactors. However, for arbitrary weighted graphs, which represent a richer variety of metric spaces, methods for finding isometric embeddings or determining their existence remain elusive, and indeed pseudofactorization and factorization have not previously been extended to this context. In this work, we address the problem of finding the factorization and pseudofactorization of a weighted graph , where satisfies the property that every edge constitutes a shortest path between its endpoints. We term such graphs minimal graphs, noting that every graph can be made minimal by removing edges not affecting its path metric. We generalize pseudofactorization and factorization to minimal graphs and develop new proof techniques that extend the previously proposed algorithms due to Graham and Winkler [Graham and Winkler, '85] and Feder [Feder, '92] for pseudofactorization and factorization of unweighted graphs. We show that any -vertex, -edge graph with positive integer edge weights can be factored in time, plus the time to find all pairs shortest paths (APSP) distances in a weighted graph, resulting in an overall running time of time. We also show that a pseudofactorization for such a graph can be computed in time, plus the time to solve APSP, resulting in an running time.

Guarding isometric subgraphs and cops and robber in planar graphs

In the game of Cops and Robbers, one of the most useful results is that an isometric path in a graph can be guarded by one cop. In this paper, we introduce the concept of wide shadow in a subgraph, and use it to characterize all 1-guardable graphs. As an application, we show that 3 cops can capture a robber in any planar graph with the added restriction that at most two cops can move simultaneously, proving a conjecture of Yang and strengthening a classical result of Aigner and Fromme.

Treelength of series–parallel graphs

The length of a tree-decomposition of a graph is the maximum distance (in the graph) between two vertices of a same bag of the decomposition. The treelength of a graph is the minimum length among its tree-decompositions. Treelength of graphs has been studied for its algorithmic applications in classical metric problems such as Traveling Salesman Problem or metric dimension of graphs and also, in compact routing in the context of distributed computing. Deciding whether the treelength of a general graph is at most 2 is NP-complete (graphs of treelength one are precisely the chordal graphs), and it is known that the treelength of a graph cannot be approximated up to a factor less than 32 (the best known approximation algorithm for treelength has an approximation ratio of 3). However, nothing is known on the computational complexity of treelength in planar graphs, except that the treelength of any outerplanar graph is equal to the third of the size of a largest isometric cycle. This work initiates the study of treelength in planar graphs by considering the next natural subclass of planar graphs, namely the one of series–parallel graphs.We first fully describe the treelength of melon graphs (set of pairwise internally disjoint paths linking two vertices), showing that, even in such a restricted graph class, the expression of the treelength is not trivial. Then, we show that treelength can be approximated up to a factor 32 in series–parallel graphs. Our main result is a quadratic-time algorithm for deciding whether a series–parallel graph has treelength at most 2. Our latter result relies on a characterization of series–parallel graphs with treelength 2 in terms of an infinite family of forbidden isometric subgraphs.

Graphs with $$G^p$$-connected medians

The median of a graph G with weighted vertices is the set of all vertices x minimizing the sum of weighted distances from x to the vertices of G. For any integer $$p\ge 2$$ , we characterize the graphs in which, with respect to any non-negative weights, median sets always induce connected subgraphs in the pth power $$G^p$$ of G. This extends some characterizations of graphs with connected medians (case $$p=1$$ ) provided by Bandelt and Chepoi (SIAM J Discrete Math 15(2):268–282, 2002. https://doi.org/10.1137/S089548019936360X ). The characteristic conditions can be tested in polynomial time for any p. We also show that several important classes of graphs in metric graph theory, including bridged graphs (and thus chordal graphs), graphs with convex balls, bucolic graphs, and bipartite absolute retracts, have $$G^2$$ -connected medians. Extending the result of Bandelt and Chepoi that basis graphs of matroids are graphs with connected medians, we characterize the isometric subgraphs of Johnson graphs and of halved-cubes with connected medians.

A Relation Between Wiener Index and Mostar Index for Daisy Cubes

Daisy cubes are a class of isometric subgraphs of the hypercubes Q n. Daisy cubes include some previously well known families of graphs like Fibonacci cubes and Lucas cubes. Moreover they appear in chemical graph theory. Two distance invariants, Wiener and Mostar indices, have been introduced in the context of the mathematical chemistry. The Wiener index W (G) is the sum of distance between over all unordered pairs of vertices of a graph G. The Mostar index Mo(G) is a measure of how far G is from being distance balanced. In this paper we establish that the Wiener and the Mostar indices of a daisy cube G are linked by the relation 2W (G) -- Mo(G) = |V (G)||E(G)|. We give also an expression of Wiener and Mostar indices for daisy cubes.

Helly-gap of a graph and vertex eccentricities

A new metric parameter for a graph, Helly-gap, is introduced. A graph G is called α-weakly-Helly if any system of pairwise intersecting disks in G has a nonempty common intersection when the radius of each disk is increased by an additive value α. The minimum α for which a graph G is α-weakly-Helly is called the Helly-gap of G and denoted by α(G). The Helly-gap of a graph G is characterized by distances in the injective hull H(G), which is a (unique) minimal Helly graph which contains G as an isometric subgraph. This characterization is used as a tool to generalize many eccentricity related results known for Helly graphs (α(G)=0), as well as for chordal graphs (α(G)≤1), distance-hereditary graphs (α(G)≤1) and δ-hyperbolic graphs (α(G)≤2δ), to all graphs, parameterized by their Helly-gap α(G). Several additional graph classes are shown to have a bounded Helly-gap, including AT-free graphs and graphs with bounded tree-length, bounded chordality or bounded αi-metric.

Treelength of Series-parallel Graphs

The length of a tree-decomposition of a graph is the maximum distance between two vertices of a same bag of the decomposition. The treelength of a graph is the minimum length among its tree-decompositions. Treelength of graphs has been studied for its algorithmic applications in classical metric problems such as Traveling Salesman Problem or metric dimension of graphs and also, in compact routing in the context of distributed computing. Deciding whether the treelength of a general graph is at most 2 is NP-complete (graphs of treelength one are precisely the chordal graphs), and it is known that the treelength of a graph cannot be approximated up to a factor less than 3/2 (the best known approximation algorithm for treelength has an approximation ratio of 3). However, nothing is known on the computational complexity of treelength in planar graphs, except that the treelength of any outerplanar graph is equal to the third of the maximum size of its isometric cycles. This work initiates the study of treelength in planar graphs by considering the next natural superclass of outerplanar graphs, namely the one of series-parallel graphs.We first fully describe the treelength of melon graphs (set of pairwise internally disjoint paths linking two vertices), showing that, even in such a restricted graph class, the expression of the treelength is not trivial. Then, we show that treelength can be approximated up to a factor 3/2 in series-parallel graphs. Our main result is a polynomial-time algorithm for deciding whether a series-parallel graph has treelength at most 2. Our latter result relies on a characterization of series-parallel graphs with treelength 2 in terms of infinite families of forbidden isometric subgraphs.

Two-Dimensional Partial Cubes

We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VC-dimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986) in a strong sense. The latter is a central conjecture of the area of computational machine learning, that is far from being solved even for general set systems of VC-dimension 2. Moreover, we point out relations to tope graphs of COMs of low rank and region graphs of pseudoline arrangements.

Circular embeddability of isometric words

Let f be a binary word and n≥1. Then the generalized Lucas cube Qn(f↽) is the graph obtained from the n-cube Qn by removing all vertices that have a circulation containing f as a factor. Ilić, Klavžar and Rho solved the question for which f and n, Qn(f↽) is an isometric subgraph of Qn for all binary words of length at most five. This question is further studied in this paper. For an isometric word f, sufficient and necessary conditions of Qn(f↽) being an isometric subgraph of Qn are found, and two problems on generalized Lucas cubes are also listed.

L1-Embeddability Under Gate-Sum Operation of Two l1-Graphs

An l 1 -graph is one in which the vertices can be labeled by binary vectors such that the Hamming distance between two binary addresses is, up to scale, the distance in the graph of corresponding vertices. This study is to determine whether the gate-sum operation can inherit the l 1 -embeddability. The subgraph H of a graph G is called a gate subgraph if, for every vertex v ∈ V (G), there exists a vertex x ∈ V (H) such that for every vertex u of H, x lies on a shortest path from v to u. The graph G is defined as the gate-sum of two graphs G 1 and G 2 with respect to H if H is a gate subgraph of at least one of G 1 and G 2 , such that G 1 ∪ G 2 = G, G 1 ∩ G 2 = H, and both G 1 and G 2 are isometric subgraphs of G. In this article, we show that the gate-sum graph of two l 1 -graphs is also an l 1 -graph.

Isometric subgraphs for Steiner distance

AbstractLet G be a connected graph and a length‐function on the edges of G. The Steiner distance sdG(A) of A ⊆ V(G) within G is the minimum length of a connected subgraph of G containing A, where the length of a subgraph is the sum of the lengths of its edges. It is clear that every subgraph H ⊆ G, endowed with the induced length‐function ℓ∣E(H), satisfies sdH(A) ≥ sdG(A) for every A ⊆ V(H). Given an integer k ≥ 2, we call H ⊆ G k‐isometric in G if equality is attained for every A ⊆ V(H) with ∣A∣ ≤ k. A subgraph is fully isometric if it is k‐isometric for every . It is easy to construct examples of graphs H ⊆ G such that H is k‐isometric, but not (k + 1)‐isometric, so this defines a strict hierarchy of properties. We are interested in situations in which this hierarchy collapses in the sense that if H ⊆ G is k‐isometric, then H is already fully isometric in G. Our first result of this kind asserts that if T is a tree and T ⊆ G is 2‐isometric with respect to some length‐function ℓ, then it is fully isometric. This fails for graphs containing a cycle. We then prove that if C is a cycle and C ⊆ G is 6‐isometric, then C is fully isometric. We present an example showing that the number 6 is indeed optimal. We then develop a structural approach toward a more general theory and present several open questions concerning the big picture underlying this phenomenon.

Colouring exact distance graphs of chordal graphs

For a graph G=(V,E) and positive integer p, the exact distance-pgraph G[♮p] is the graph with vertex set V and with an edge between vertices x and y if and only if x and y have distance p. Recently, there has been an effort to obtain bounds on the chromatic number χ(G[♮p]) of exact distance-p graphs for G from certain classes of graphs. In particular, if a graph G has tree-width t, it has been shown that χ(G[♮p])∈O(pt−1) for odd p, and χ(G[♮p])∈O(ptΔ(G)) for even p. We show that if G is chordal and has tree-width t, then χ(G[♮p])∈O(pt2) for odd p, and χ(G[♮p])∈O(pt2Δ(G)) for even p.If we could show that for every graph H of tree-width t there is a chordal graph G of tree-width t which contains H as an isometric subgraph (i.e., a distance preserving subgraph), then our results would extend to all graphs of tree-width t. While we cannot do this, we show that for every graph H of genus g there is a graph G which is a triangulation of genus g and contains H as an isometric subgraph.

On a conjecture about the existence of isometric subgraphs

AbstractIt has been conjectured that for an ‐vertex graph , if , then has at least one isometric subgraph of every possible order. On the other hand, it is known that , for any , does not guarantee that has this property. A graph having an isometric subgraph of every order will be called distance preserving graph. The best bound in the literature is , which guarantees that the graph is distance preserving. In this paper, using a combinatorial method, it is proved that for every there is an integer such that every graph with vertices and is distance preserving.

Daisy cubes: A characterization and a generalization

Daisy cubes are a recently introduced class of isometric subgraphs of hypercubes Qn. They are induced with intervals between chosen vertices of Qn and the vertex 0n∈V(Qn). In this paper we characterize daisy cubes in terms of an expansion procedure thus answering an open problem proposed by Klavžar and Mollard, 2019, in the introductory paper of daisy cubes (Klavžar and Mollard, 2019). To obtain such a characterization several interesting properties of daisy cubes are presented. For a given graph G isomorphic to a daisy cube, but without the corresponding embedding into a hypercube, we present an algorithm which finds a proper embedding of G into a hypercube in O(mn) time. Finally, daisy graphs of a rooted graph are introduced and shown to be a generalization of daisy cubes.

Hypercellular graphs: Partial cubes without [formula omitted] as partial cube minor

We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex Q3− (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph can be obtained from the Cartesian products of edges and even cycles via successive gated amalgams. This decomposition result enables us to establish a variety of results. In particular, it yields that our class of graphs generalizes median and cellular graphs, which motivates naming our graphs hypercellular. Furthermore, we show that hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids. Finally, we characterize hypercellular graphs as being median-cell — a property naturally generalizing the notion of median graphs.

Bundling all shortest paths

We study the problem of finding a minimum bundling set in a graph, where a bundling set is a set B of vertices such that every shortest path can be extended to a shortest path from a vertex in B to some other vertex. If G is a weighted graph, we denote the size of a minimum bundling set in G by b(G).Bundling sets can be used by the ALT algorithm that finds shortest paths in weighted graphs. For a fixed bundling set B in a weighted graph G, after some preprocessing using O(|B||V(G)|) memory, it is possible to determine the distance between any two vertices in time O(|B|). Therefore, it is desirable to find small bundling sets.We show that determining b(G) is NP-hard and give a 2-approximation algorithm. Moreover we characterize simple graphs with b=1 and subgraphs of grids with b=2.We also introduce the parameter b∗(G) equal to the minimum of b(H) over all weighted graphs H such that G is an isometric subgraph of H, i.e. for every two vertices u,v of G the distances from u to v in G and in H are the same. Sometimes b∗(G) is much smaller than b(G) and a further improvement of performance of route planning can be obtained.As a part of a proof, we show that at least Θlognloglogn triangle-free graphs are needed to cover a complete graph on n vertices, which may be of independent interest.

Computing metric hulls in graphs

We prove that, given a closure function the smallest preimage of a closed set can be calculated in polynomial time in the number of closed sets. This implies that there is a polynomial time algorithm to compute the convex hull number of a graph, when all its convex subgraphs are given as input. We then show that deciding if the smallest preimage of a closed set is logarithmic in the size of the ground set is LOGSNP-hard if only the ground set is given. A special instance of this problem is to compute the dimension of a poset given its linear extension graph, that is conjectured to be in P. The intent to show that the latter problem is LOGSNP-complete leads to several interesting questions and to the definition of the isometric hull, i.e., a smallest isometric subgraph containing a given set of vertices $S$. While for $|S|=2$ an isometric hull is just a shortest path, we show that computing the isometric hull of a set of vertices is NP-complete even if $|S|=3$. Finally, we consider the problem of computing the isometric hull number of a graph and show that computing it is $\Sigma^P_2$ complete.

Modular decomposition of graphs and the distance preserving property

Given a graph G, a subgraph H is isometric if dH(u,v)=dG(u,v) for every pair u,v∈V(H), where d is the distance function. A graph G is distance preserving (dp) if it has an isometric subgraph of every possible order. A graph is sequentially distance preserving (sdp) if its vertices can be ordered such that deleting the first i vertices results in an isometric subgraph, for all i≥1. We introduce a generalisation of the lexicographic product of graphs, which can be used to non-trivially describe graphs. This generalisation is the inverse of the modular decomposition of graphs, which divides the graph into disjoint clusters called modules. Using these operations, we give a necessary and sufficient condition for graphs to be dp. Finally, we show that the Cartesian product of a dp graph and an sdp graph is dp.

Cube-complements of generalized Fibonacci cubes

The generalized Fibonacci cube Qh(f) is the graph obtained from the h-cube Qh by removing all vertices that contain a given binary string f as a substring. If G is an induced subgraph of Qh, then the cube-complement of G is the graph induced by the vertices of Qh which are not in G. In particular, the cube-complement of a generalized Fibonacci cube Qh(f) is the subgraph of Qh induced by the set of all vertices that contain f as a substring. The questions whether a cube-complement of a generalized Fibonacci cube is (i) connected, (ii) an isometric subgraph of a hypercube or (iii) a median graph are studied. Questions (ii) and (iii) are completely solved, i.e. the sets of binary strings that allow a graph of this class to be an isometric subgraph of a hypercube or a median graph are given. The cube-complement of a daisy cube is also studied.