On the Graph Isomorphism Completeness of Directed and Multidirected Graphs

In his pioneering paper on matroids in 1935, Whitney obtained a characterization for binary matroids and left a comment at end of the paper that the problem of characterizing graphic matroids is the same as that of characterizing matroids which correspond to matrices (mod 2) with exactly two ones in each column. Later on Tutte obtained a characterization of graphic matroids in terms of forbidden minors in 1959. It is clear that Whitney indicated about incidence matrices of simple undirected graphs. Here we introduce the concept of a segment binary matroid which corresponds to matrices over $\mathbb{Z}_2$ which has the consecutive $1$'s property (i.e., $1$'s are consecutive) for columns and obtained a characterization of graphic matroids in terms of this. In fact, we introduce a new representation of simple undirected graphs in terms of some vectors of finite dimensional vector spaces over $\mathbb{Z}_2$ which satisfy consecutive $1$'s property. The set of such vectors is called a coding sequence of a graph $G$. Among all such coding sequences we identify the one which is unique for a class of isomorphic graphs. We call it the code of the graph. We characterize several classes of graphs in terms of coding sequences. It is shown that a graph $G$ with $n$ vertices is a tree if and only if any coding sequence of $G$ is a basis of the vector space $\mathbb{Z}_2^{n-1}$ over $\mathbb{Z}_2$. Moreover considering coding sequences as binary matroids, we obtain a characterization for simple graphic matroids and found a necessary and sufficient condition for graph isomorphism in terms of a special matroid isomorphism between their corresponding coding sequences. For this, we introduce the concept of strong isomorphisms of segment binary matroids and show that two simple (undirected) graphs are isomorphic if and only if their canonical sequences are strongly isomorphic segment binary matroids.

Research on space efficient graph algorithms, particularly for st-connectivity, has a long history including the celebrated polynomial time, O(lg n) bits1 algorithm in undirected graphs by Reingold J. JACM. 55(4) (2008), and polynomial time, \(n/2^{\Theta (\sqrt {\lg n})}\) bits algorithm in directed graphs by Barnes et al. SICOMP. 27(5), 1273–1282 (1998). Recent works by Asano et al. ISAAC (2014) and Elmasry et al. STACS (2015), reconsidered classical fundamental graph algorithms focusing on improving the space complexity. Elmasry et al. gave, among others, an implementation of breadth first search (BFS) in a graph G with n vertices and m edges, taking the optimal O(m + n) time using O(n) bits improving the naive O(n lg n) bits implementation. Similarly, Asano et al. provided space efficient implementations for performing depth first search (DFS) in a graph G. We continue this line of work focusing on improving the space requirement for implementing a few classical graph algorithms. Our first result is a simple data structure that can maintain any subset S of a universe of u elements using just u + o(u) bits and supports in constant time, apart from the standard insert, delete and membership queries, the operation findany that finds and returns any element of the set (or outputs that the set is empty). It can also enumerate all elements present currently in the set in no particular order in O(k + 1) time where k is the number of elements currently belonging to the set. While this data structure supports a weaker set of operations than that of Elmasry et al. STACS (2015), it is simple, more space efficient and is sufficient to support a BFS implementation optimally in O(m + n) time using at most 2n + o(n) bits. Later, we further improve the space requirement of BFS to at most n lg 3 + o(n) bits albeit with a slight increase in running time to O(m lg nf(n)) time where f(n) is any extremely slow growing function of n, and the o term in the space is a function of f(n). We also discuss a similar time-space tradeoff result for finding a minimum weight spanning tree of a weighted (bounded by polynomial in n) undirected graph using n + O(n/f(n)) bits and O(m lg nf(n)) time, for any function f(n) such that 1 ≤ f(n) ≤ n. For DFS in a graph G, we provide an implementation taking O(m + n) time and O(n lg m/n) bits. This partially answers at least for sparse graphs, a question asked by Asano et al. ISAAC (2014) whether DFS can be performed in O (m + n) time and using O(n) bits, and also simultaneously improves the DFS result of Elmasry et al. STACS (2015). Using our DFS algorithm and other careful implementations, we show how one can also test for biconnectivity, 2-edge connectivity, and find cut vertices and bridges of a given undirected graph within the same time and space bounds; earlier classical linear time algorithms for these problems used Ω(n lg n) bits of space.

For an undirected graph G¿=¿(V,E), we say that for l,u,v¿¿¿V, l separates u from v if the distance between u and l differs from the distance from v to l. A set of vertices L¿¿¿V is a feasible solution if for every pair of vertices u,v¿¿¿V there is l¿¿¿L that separates u from v. The metric dimension of a graph is the minimum cardinality of such a feasible solution. Here, we extend this well-studied problem to the case where each vertex v has a non-negative cost, and the goal is to find a feasible solution with a minimum total cost. This weighted version is NP-hard since the unweighted variant is known to be NP-hard. We show polynomial time algorithms for the cases where G is a path, a tree, a cycle, a cograph, a k-edge-augmented tree (that is, a tree with additional k edges) for a constant value of k, and a (not necessarily complete) wheel. The results for paths, trees, cycles, and complete wheels extend known polynomial time algorithms for the unweighted version, whereas the other results are the first known polynomial time algorithms for these classes of graphs even for the unweighted version. Next, we extend the set of graph classes for which computing the unweighted metric dimension of a graph is known to be NPC by showing that for split graphs, bipartite graphs, co-bipartite graphs, and line graphs of bipartite graphs, the problem of computing the unweighted metric dimension of the graph is NPC.

Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances

A locating-dominating set of an undirected graph is a subset of vertices S such that S is dominating and for every u,v∉S, the neighbourhood of u and v on S are distinct (i.e. N(u)∩S≠N(v)∩S). Locating-dominating sets have received a considerable attention in the last decades. In this paper, we consider the oriented version of the problem. A locating-dominating set in an oriented graph is a set S such that for each w∈V∖S, N−(w)∩S≠∅ and for each pair of distinct vertices u,v∈V∖S, N−(u)∩S≠N−(v)∩S. We consider the following two parameters. Given an undirected graph G, we look for γ→LD(G) (Γ→LD(G)) which is the size of the smallest (largest) optimal locating-dominating set over all orientations of G. In particular, if D is an orientation of G, then γ→LD(G)≤γLD(D)≤Γ→LD(G) where γLD(D) is the minimum size of a locating-dominating set of D.For the best orientation, we prove that, for every twin-free graph G on n vertices, γ→LD(G)≤n/2 which proves a “directed version” of a widely studied conjecture on the location-domination number. As a side result we obtain a new improved upper bound for the location-domination number in undirected trees. Moreover, we give some bounds for γ→LD(G) on many graph classes and drastically improve the value n/2 for (almost) d-regular graphs by showing that γ→LD(G)∈O(log⁡d/d⋅n) using a probabilistic argument.While γ→LD(G)≤γLD(G) holds for every graph G, we give some graph classes such as outerplanar graphs for which Γ→LD(G)≥γLD(G) and some for which Γ→LD(G)≤γLD(G) such as complete graphs. We also give general bounds for Γ→LD(G) such as Γ→LD(G)≥α(G). Finally, we show that for many graph classes Γ→LD(G) is polynomial on n but we leave open the question whether there exist graphs with Γ→LD(G)∈O(log⁡n).

Matching interdiction

A graph with a vertex set is said to be a prime graph if there exists a bijective mapping , where denotes the number of vertices in , such that for any two adjacent vertices and in have . Tensor Product graph is a way to combine (compose) two graphs into one larger and more complex graph. The result is a new graph that reflects the connection properties of the two original graphs, but in a very specific and more complex way than other graph operations. Therefore, this research aims to determine whether there is prime labeling in the class of graphs resulting from the Tensor Product of the path graph and the cycle graph . The research employed analytical and exploratory methods with a trial-and-error strategy to determine the labeling that possesses a prime property. The results of this study prove that two classes of the Tensor Product graph for , and graph , for are prime graph. This finding expands the results on classes of graphs that admit prime labeling and provides a basis for further research on graph labeling in other graph operations

For any class C of bipartite graphs, we define quasi-C to be the class of all graphs G such that every bipartition of G belongs to C. This definition is motivated by a generalisation of the switch Markov chain on perfect matchings from bipartite graphs to nonbipartite graphs. The monotone graphs, also known as bipartite permutation graphs and proper interval bigraphs, are such a class of bipartite graphs. We investigate the structure of quasi-monotone graphs and hence construct a polynomial time recognition algorithm for graphs in this class.

Similarity is an important way of categorizing mathematical objects. In this paper we consider how similarity between undirected graphs, or networks, can be defined. We do this by reducing any undirected graph, G = (V,E) to its irreducible spine, I. First we show that the reduction process, ω, is a unique transformation and thus is a well-defined function. Each irreducible spine, I, thus defines an equivalence class consisting of all graphs {G_i} such that G_i.ω = I. We then show that reduction by ω preserves many key properties. Specifically it preserves shortest path lengths in I, and (under rather general conditions) preserves the location of distance and betweeness centers. In essence, every member graph of the equivalence class will have much the same path structure. Next, an algorithm is presented which, given any irreducible spine I, will generate (randomly) a graph Gk in the equivalence class defined by I, that is, we must have G_k.ω = I. Since I.e.ω = I, e = ω^-1. It is of some importance in network analysis to be able to quantify the notion of network similarity; to be able to say that a network G1 is closer to the network G2 than to G3. A popular method of comparing similarity between two networks G1 and G2 is to calculate the Spearman coefficient based on corresponding node degrees. We do this for 5 random graphs in the equivalence class of our running example. Finally, we show that any irreducible spine consists of a system of cycles. We then note that every spine, I, is topologically equivalent to a cubic graph. Thus individual cubic graphs generate a second, and coarser, equivalence relation on the family of all undirected graphs, which we call its genus.

Computing a minimum outer-connected dominating set for the class of chordal graphs

In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time. An interesting family of graph classes arises from intersection graphs of geometric objects. In this work we show that the Graph Isomorphism Problem for unit square graphs, intersection graphs of axis-parallel unit squares in the plane, can be solved in polynomial time. Since the recognition problem for this class of graphs is NP-hard we can not rely on standard techniques for geometric graphs based on constructing a canonical realization. Instead, we develop new techniques which combine structural insights into the class of unit square graphs with understanding of the automorphism group of such graphs. For the latter we introduce a generalization of bounded degree graphs which is used to capture the main structure of unit square graphs. Using group theoretic algorithms we obtain sufficient information to solve the isomorphism problem for unit square graphs.

We consider the approximability of the maximum edge-disjoint paths problem (MEDP) in undirected graphs, and in particular, the integrality gap of the natural multicommodity flow based relaxation for it. The integrality gap is known to be \(\Omega(\sqrt{n})\) even for planar graphs [11] due to a simple topological obstruction and a major focus, following earlier work [14], has been understanding the gap if some constant congestion is allowed. In planar graphs the integrality gap is O(1) with congestion 2 [19,5]. In general graphs, recent work has shown the gap to be O(polylog(n)) [8,9] with congestion 2. Moreover, the gap is Ω(logΩ(c) n) in general graphs with congestion c for any constant c ≥ 1 [1].It is natural to ask for which classes of graphs does a constant-factor constant-congestion property hold. It is easy to deduce that for given constant bounds on the approximation and congestion, the class of “nice” graphs is minor-closed. Is the converse true? Does every proper minor-closed family of graphs exhibit a constant-factor constant-congestion bound relative to the LP relaxation? We conjecture that the answer is yes. One stumbling block has been that such bounds were not known for bounded treewidth graphs (or even treewidth 3). In this paper we give a polytime algorithm which takes a fractional routing solution in a graph of bounded treewidth and is able to integrally route a constant fraction of the LP solution’s value. Note that we do not incur any edge congestion. Previously this was not known even for series parallel graphs which have treewidth 2. The algorithm is based on a more general argument that applies to k-sums of graphs in some graph family, as long as the graph family has a constant-factor constant-congestion bound. We then use this to show that such bounds hold for the class of k-sums of bounded genus graphs.KeywordsPlanar GraphTree DecompositionGraph FamilyDisjoint Path ProblemSeries Parallel GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

An odd dominating set of a simple, undirected graph G = (V, E) is a set of vertices D ⊆ V such that |N [v]∩D| ≡ 1 mod 2 for all vertices v ∈ V . It is known that every graph has an odd dominating set. In this paper we consider the concept of connected odd dominating sets. We prove that the problem of deciding if a graph has a connected odd dominating set is NP-complete. We also determine the existence or non-existence of such sets in several classes of graphs. Among other results, we prove there are only 15 grid graphs that have a connected odd dominating set. AMS subject classification index: primary 05C35.

Fully Dynamic Algorithms for Maintaining Shortest Paths Trees

Given an undirected graph G = (V, E) with n vertices and a positive length w(e) on each edge e ∈ E, we consider Minimum Average Distance (MAD) spanning trees i.e., trees that minimize the path length summed over all pairs of vertices. One of the first results on this problem is due to Wong who showed in 1980 that a Distance Preserving (DP) spanning tree rooted at the median of G is a 2-approximate solution. On the other hand, Dankelmann has exhibited in 2000 a class of graphs where no MAD spanning tree is distance preserving from a vertex. We establish here a new relation between MAD and DP trees in the particular case where the lengths are integers. We show that in a MAD spanning tree of G, each subtree H′ = (V′, E′) consisting of a vertex [Formula: see text] and the union of branches of [Formula: see text] that are each of size less than or equal to [Formula: see text], where w+ is the maximum edge-length in G, is a distance preserving spanning tree of the subgraph of G induced by V′.