Going Far from Degeneracy

The thesis consists of three chapters. The first chapter introduces the basic notions of graph theory and defines vertex-reconstruction and edge-reconstruction problem. The second chapter and third chapter are devoted to the edge-reconstruction of bi-degreed graphs and bipartite graphs respectively. A bi-degreed graph G is a graph with two degrees d > δ. By elementary arguments we can assume d = δ + 1 and there are at least two vertices of degree δ. Call vertices of degree d "big" vertex and degree δ "small" vertex. Define "symmetric" path of length p Sp to be one with both ends small vertices and all other internal vertices big vertices; define "asymmetric" path of length p Ap to be one with one end a small vertex and all others big vertices. If s(G) is the minimum distance between two small vertices in G, we can show that s(G) is "independent" of G (i.e. it is edge-reconstructable), and that G has at most one nonisomorphic edge-reconstruction H. From this, the concept of "forced move" posed by Dr. Swart is obvious. Using the principle of forced move (and sometimes also "forced edge" posed by Dr. Swart as well), it's easy to derive a few interesting properties, like say G is edge-reconstructable if s(G) is even or if two Ss(G)'s intersect at an internal vertex, etc. Write s for s(G). When s is odd, consider the concept of s - n-chain, which is n Ss's following from end to end. We can show first s - 3-chain and then s - 2-chain cannot exist. Hence Ss's are disjoint. Think of Ss's as "lines" in some geometry. Define two more "distance" functions s1 and s2 such that s1 "represents" the distance from a point to a line and s2 means the distance between to "skew" lines. With the aid of forced move principle again, we can at last prove every bi-degreed graph with at least four edges is edge-reconstructable. A bipartite graph G is a graph whose vertex set V is the disjoint union of two sets v1 and v2 such that every edge joins v1 and v2. By elementary reduction we can assume G to be connected. We define special chains inductively so that it starts at a vertex of minimum degree and always goes to a neighbor or minimum degree. Special chains will be the main tool to prove edge-reconstructability. By G's finiteness, we note they will "terminate" somewhere, and we have three types of termination for them. Let condition A•s be that degree sequence of special chain is edge-reconstructable, condition Bi's be that number of special chains is edge-reconstructable(and some more general variations); condition P's be that the "last vertices" of two special chains be not adjacent; we can prove that all A, Bi and P's should hold inductively in an interlocked way. (This is a big task). Then condition P's can be used to prove G's edge-reconstructability for all three types of termination. We can then prove every bipartite graph with at least four edges is edge-reconstructable.

We consider the parameterized problem of counting all matchings with exactly k edges in a given input graph G. This problem is #W[1]-hard (Curticapean, ICALP 2013), so it is unlikely to admit f(k)poly(n) time algorithms. We show that #W[1]-hardness persists even when the input graph G comes from restricted graph classes, such as line graphs and bipartite graphs of arbitrary constant girth and maximum degree two on one side. To prove the result for line graphs, we observe that k-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of k paths of length two into (arbitrary) host graphs. Here, a homomorphism from H to G is edge-injective if it maps any two distinct edges of H to distinct edges in G. We show that edge-injective homomorphisms from a pattern graph H can be counted in polynomial time if H has bounded vertex-cover number after removing isolated edges. For hereditary classes H of pattern graphs, we obtain a full complexity dichotomy theorem by proving that counting edge-injective homomorphisms, restricted to patterns from H, is #W[1]-hard if no such bound exists. Our proofs rely on an edge-colored variant of Holant problems and a delicate interpolation argument; both may be of independent interest.

Chordal graphs are the intersection graphs of subtrees of a tree, while interval graphs of subpaths of a path. Undirected path graphs, directed path graphs and rooted directed path graphs are intermediate graph classes, defined, respectively, as the intersection graphs of paths of a tree, of directed paths of an oriented tree, and of directed paths of an out branching. All of these path graphs have vertex leafage 2. Dominating Set, Connected Dominating Set, and Steiner tree problems are ‐hard parameterized by the size of the solution on chordal graphs, ‐complete on undirected path graphs, and polynomial‐time solvable on rooted directed path graphs, and hence also on interval graphs. We further investigate the (parameterized) complexity of all these problems when constrained to chordal graphs, taking the vertex leafage and the aforementioned classes into consideration. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are on chordal graphs when parameterized by the size of the solution plus the vertex leafage, and that Weighted Connected Dominating Set is polynomial‐time solvable on strongly chordal graphs. We also introduce a new subclass of undirected path graphs, which we call in–out rooted directed path graphs, as the intersection graphs of directed paths of an in–out branching. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are solvable in polynomial time on this class, generalizing the polynomiality for rooted directed path graphs proved by Booth and Johnson (SIAM J. Comput. 11 (1982), 191‐199.) and by White et al. (Networks 15 (1985), 109‐124.).

In this paper we revisit the classical Edge Disjoint Paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or fpt) algorithms. As our first result, we answer an open question stated in Fleszar, Mnich, and Spoerhase (2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain NP-hard even for treewidth two, a result by Zhou et al. (2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an fpt-algorithm has remained open since then. We show that this is highly unlikely by establishing the W[1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an fpt-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.

Relative lengths of paths and cycles in 3-connected graphs

Motivated by challenges related to domination, connectivity, and information propagation in social and other networks, we initiate the study of the VECTOR CONNECTIVITY problem. This problem takes as input a graph G and an integer kv for every vertex v of G, and the objective is to find a vertex subset S of minimum cardinality such that every vertex v either belongs to S, or is connected to at least kv vertices of S by disjoint paths. If we require each path to be of length exactly 1, we get the well‐known VECTOR DOMINATION problem, which is a generalization of the famous DOMINATING SET problem and several of its variants. Consequently, our problem becomes NP‐hard if an upper bound on the length of the disjoint paths is also supplied as input. Due to the hardness of these domination variants even on restricted graph classes, like split graphs, VECTOR CONNECTIVITY seems to be a natural problem to study for drawing the boundaries of tractability for this type of problems. We show that VECTOR CONNECTIVITY can actually be solved in polynomial time on split graphs, in addition to cographs and trees. We also show that the problem can be approximated in polynomial time within a factor of on all n‐vertex graphs.Copyright © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 63(4), 277–285 2014

A faster algorithm to recognize undirected path graphs

In the article, the definition of an undirected multiple graph of any natural multiplicity $$k > 1$$ is stated. There are edges of three types: ordinary edges, multiple edges, and multi-edges. Each edge of the last two types is the union of $$k$$ linked edges, which connect 2 or $$k + 1$$ vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges and it can be the common ending vertex to $$k$$ linked edges of some multi-edge. If a vertex is the common end of a multi-edge, it cannot be the common end of any other multi-edge. Also, a class of the divisible multiple graphs is considered. The main peculiarity of them is a possibility to divide the graph into $$k$$ parts, which are adjusted on the linked edges and which have no common ordinary edges. Each part is an ordinary graph. The following terms are generalized: the degree of a vertex, connectedness of a graph, the path, the cycle, the weight of an edge, and the path length. The definition of a reachability set for the ordinary and multiple edges is stated. The adjacency property is defined for a pair of reachability sets. It is shown, that we can check the connectedness of a multiple graph with the polynomial algorithm based on the search for reachability sets and testing their adjacency. A criterion of the existence of a multiple path between two given vertices is considered. The shortest multiple path problem is stated. Then, we suggest an algorithm for finding the shortest path in a multiple graph. It uses Dijkstra’s algorithm for finding the shortest paths in subgraphs, which correspond to different reachability sets.

A heuristic search algorithm is given that determines and resolves the Hamiltonian circuit problem in directed graphs. The heuristic information of each vertex is a set composed of its possible path length values from the starting vertex, which is obtained by the path length extension algorithm. A detailed analysis of algorithm examples shows that the heuristic algorithm can greatly reduce the number of processing nodes compared to the backtracking algorithm without information; meanwhile, the algorithm’s time complexity is related to the number of loops in the directed graph. If a directed graph G(V, E) only contains loops that contain the starting point, then the Hamiltonian problem of G can be determined in polynomial time. The algorithm is implemented using C++ and Python programming, and all of the test graphs are generated by the random graph algorithm. The experiments show that the heuristic algorithm can reduce the number of extended nodes by at least 10% in directed random graphs with no or only a few loops. This heuristic algorithm can save time in applications that perform tasks while extending nodes because it reduce the number of extended nodes.

Given a graph G, and terminal vertices s and t, the Tracking Paths problem asks to compute a set of minimum number of vertices to be marked as trackers, such that the sequence of trackers encountered in each s-t path is unique. Tracking Paths is NP-hard in both directed and undirected graphs in general. In this paper we give a collection of polynomial time algorithms for some restricted versions of Tracking Paths. We prove that Tracking Paths is polynomial time solvable for undirected chordal graphs and tournament graphs. We also show that Tracking Paths is NP-hard in graphs with bounded maximum degree Delta ge 6, and give a 2(Delta +1)-approximate algorithm for this case. Further, we give a polynomial time algorithm which, given an undirected graph G, a tracking set Tsubseteq V(G), and a sequence of trackers pi , returns the unique s-t path in G that corresponds to pi , if one exists. Finally we analyze the version of tracking s-t paths where paths are tracked using edges instead of vertices, and we give a polynomial time algorithm for the same.

Hamiltonian cycles and paths in vertex-transitive graphs with abelian and nilpotent groups

A class of [formula omitted]-perfect graphs

We consider the problem of estimating the size of a maximum matching when the edges are revealed in a streaming fashion. When the input graph is planar, we present a simple and elegant streaming algorithm that, with high probability, estimates the size of a maximum matching within a constant factor using Õ( n 2/3 ) space, where n is the number of vertices. The approach generalizes to the family of graphs that have bounded arboricity, which include graphs with an excluded constant-size minor. To the best of our knowledge, this is the first result for estimating the size of a maximum matching in the adversarial-order streaming model (as opposed to the random-order streaming model) in o ( n ) space. We circumvent the barriers inherent in the adversarial-order model by exploiting several structural properties of planar graphs, and more generally, graphs with bounded arboricity. We further reduce the required memory size to Õ(√ n ) for three restricted settings: (i) when the input graph is a forest; (ii) when we have 2-passes and the input graph has bounded arboricity; and (iii) when the edges arrive in random order and the input graph has bounded arboricity. Finally, we design a reduction from the Boolean Hidden Matching Problem to show that there is no randomized streaming algorithm that estimates the size of the maximum matching to within a factor better than 3/2 and uses only o ( n 1/2 ) bits of space. Using the same reduction, we show that there is no deterministic algorithm that computes this kind of estimate in o ( n ) bits of space. The lower bounds hold even for graphs that are collections of paths of constant length.

Given an undirected graph G and a positive integer k, the k-vertex-connectivity augmentation problem is to find a smallest set F of new edges for which G+F is k-vertex-connected. Polynomial algorithms for this problem have been found only for k ≤ 4 and a major open question in graph connectivity is whether this problem is solvable in polynomial time in general.In this paper we develop the first algorithm which delivers an optimal solution in polynomial time for every fixed k. In the case when the size of an optimal solution is large compared to k, we also give a min-max formula for the size of a smallest augmenting set.A key step in our proofs is a complete solution of the augmentation problem for a new family of graphs which we call k-independence free graphs. We also prove new splitting off theorems for vertex connectivity.

General vertex disjoint paths in series-parallel graphs