Characterizing path graphs by forbidden induced subgraphs

A path graph is the intersection graph of subpaths of a tree. In 1970, Renz asked for a characterization of path graphs by forbidden induced subgraphs. We answer this question by determining the complete list of graphs that are not path graphs and are minimal with this property. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 369–384, 2009     

Rooted directed path graphs are leaf powers

Leaf powers are a graph class which has been introduced to model the problem of reconstructing phylogenetic trees. A graph G=(V,E) is called k-leaf power if it admits a k-leaf root, i.e., a tree T with leaves V such that uv is an edge in G if and only if the distance between u and v in T is at most k. Moroever, a graph is simply called leaf power if it is a k-leaf power for some k∈N. This paper characterizes leaf powers in terms of their relation to several other known graph classes. It also addresses the problem of deciding whether a given graph is a k-leaf power.We show that the class of leaf powers coincides with fixed tolerance NeST graphs, a well-known graph class with absolutely different motivations. After this, we provide the largest currently known proper subclass of leaf powers, i.e, the class of rooted directed path graphs.Subsequently, we study the leaf rank problem, the algorithmic challenge of determining the minimum k for which a given graph is a k-leaf power. Firstly, we give a lower bound on the leaf rank of a graph in terms of the complexity of its separators. Secondly, we use this measure to show that the leaf rank is unbounded on both the class of ptolemaic and the class of unit interval graphs. Finally, we provide efficient algorithms to compute 2|V|-leaf roots for given ptolemaic or (unit) interval graphs G=(V,E).     

Random dense bipartite graphs and directed graphs with specified degrees

Let s and t be vectors of positive integers with the same sum. We study the uniform distribution on the space of simple bipartite graphs with degree sequence s in one part and t in the other; equivalently, binary matrices with row sums s and column sums t . In particular, we find precise formulae for the probabilities that a given bipartite graph is edge‐disjoint from, a subgraph of, or an induced subgraph of a random graph in the class. We also give similar formulae for the uniform distribution on the set of simple directed graphs with out‐degrees s and in‐degrees t . In each case, the graphs or digraphs are required to be sufficiently dense, with the degrees varying within certain limits, and the subgraphs are required to be sufficiently sparse. Previous results were restricted to spaces of sparse graphs. Our theorems are based on an enumeration of bipartite graphs avoiding a given set of edges, proved by multidimensional complex integration. As a sample application, we determine the expected permanent of a random binary matrix with row sums s and column sums t . © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Signed domination numbers of directed graphs

The concept of signed domination number of an undirected graph (introduced by J. E. Dunbar, S. T. Hedetniemi, M. A. Henning and P. J. Slater) is transferred to directed graphs. Exact values are found for particular types of tournaments. It is proved that for digraphs with a directed Hamiltonian cycle the signed domination number may be arbitrarily small.     

Classification of reconfiguration graphs of shortest path graphs with no induced 4-cycles

For any graph $G$ with $a,b\in V\left(G\right)$, a shortest path reconfiguration graph can be formed with respect to $a$ and $b$; we denote such a graph as $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths from $a$ to $b$ in $G$ while two vertices $U,W$ in $V\left(S(G,a,b)\right)$ are adjacent if and only if the vertex sets of the paths that represent $U$ and $W$ differ in exactly one vertex. In a recent paper (Asplund et al., 2018), it was shown that shortest path graphs with girth five or greater are exactly disjoint unions of even cycles and paths. In this paper, we extend this result by classifying all shortest path graphs with no induced 4-cycles.     

Enumeration of labelled chain graphs and labelled essential directed acyclic graphs

A chain graph is a digraph whose strong components are undirected graphs and a directed acyclic graph (ADG or DAG) G is essential if the Markov equivalence class of G consists of only one element. We provide recurrence relations for counting labelled chain graphs by the number of chain components and vertices; labelled essential DAGs by the number of vertices. The second one is a lower bound for the number of labelled essential graphs. The formula for labelled chain graphs can be extended in such a way, that allows us to count digraphs with two additional properties, which essential graphs have.     

Regular path decompositions of odd regular graphs

Kotzig asked in 1979 what are necessary and sufficient conditions for a d‐regular simple graph to admit a decomposition into paths of length d for odd d>3. For cubic graphs, the existence of a 1‐factor is both necessary and sufficient. Even more, each 1‐factor is extendable to a decomposition of the graph into paths of length 3 where the middle edges of the paths coincide with the 1‐factor. We conjecture that existence of a 1‐factor is indeed a sufficient condition for Kotzig's problem. For general odd regular graphs, most 1‐factors appear to be extendable and we show that for the family of simple 5‐regular graphs with no cycles of length 4, all 1‐factors are extendable. However, for d>3 we found infinite families of d‐regular simple graphs with non‐extendable 1‐factors. Few authors have studied the decompositions of general regular graphs. We present examples and open problems; in particular, we conjecture that in planar 5‐regular graphs all 1‐factors are extendable. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 114–128, 2010     

Obstructions to chordal circular-arc graphs of small independence number

A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any two vertices of the quadruple either miss (have no neighbours on) some path connecting the remaining two vertices of the quadruple, or are connected by some path missed by the remaining two vertices. This is akin to the notion of asteroidal triple used in the classical characterization of interval graphs by Lekkerkerker and Boland [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.].In this note, we first observe that blocking quadruples are obstructions for circular-arc graphs. We then focus on chordal graphs, and study the relationship between the structure of chordal graphs and the presence/absence of blocking quadruples.Our contribution is two-fold. Firstly, we provide a forbidden induced subgraph characterization of chordal graphs without blocking quadruples. In particular, we observe that all the forbidden subgraphs are variants of the subgraphs forbidden for interval graphs [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.]. Secondly, we show that the absence of blocking quadruples is sufficient to guarantee that a chordal graph with no independent set of size five is a circular-arc graph. In our proof we use a novel geometric approach, constructing a circular-arc representation by traversing around a carefully chosen clique tree.     

On the fastest moving off from a vertex in directed regular graphs

Let Γ be a directed regular locally finite graph, and let $\bar \Gamma $ be the undirected graph obtained by forgetting the orientation of Γ. Let x be a vertex of Γ and let n be a nonnegative integer. We study the length of the shortest directed path in Γ starting at x and ending outside of the ball of radius n centered at x in $\bar \Gamma $ .     

A bound on judicious bipartitions of directed graphs

Judicious partitioning problems on graphs ask for partitions that bound several quantities simultaneously,which have received much attention lately.Scott(2005)asked the following natural question:What is the maximum constant cdsuch that every directed graph D with m arcs and minimum outdegree d admits a bipartition V(D)=V_1∪V_2 satisfying min{e(V_1,V_2),e(V_2,V_1)}cdm?Here,for i=1,2,e(V_i,V_3-i)denotes the number of arcs in D from V_i to V_3-i.Lee et al.(2016)conjectured that every directed graph D with m arcs and minimum outdegree at least d 2 admits a bipartition V(D)=V_1∪V_2 such that min{e(V_1,V_2),e(V_2,V_1)}≥((d-1)/(2(2 d-1))+o(1))m.In this paper,we show that this conjecture holds under the additional natural condition that the minimum indegree is also at least d.     

路可扩图的一个充分条件

通过讨论图中任意一对不相邻顶点的度和,对路可扩图的充分条件进行研究,得到了如下结果：设图G的阶是n,如果G中任意一对不相邻顶点的度和至少为3/2n-1,则图G是路可扩的.并且说明了这里两不相邻顶点的度和的下界3/2n-1是最好可能的.     

Directed 3-cycle decompositions of complete directed graphs with quadratic leaves

Let G be a vertex-disjoint union of directed cycles in the complete directed graph D_t, let |E(G)| be the number of directed edges of G and suppose or if t=5, and if t=6. It is proved in this paper that for each positive integer t, there exist -decompositions for D_t−G if and only if .     

The outflow ranking method for weighted directed graphs

A ranking method assigns to every weighted directed graph a (weak) ordering of the nodes. In this paper we axiomatize the ranking method that ranks the nodes according to their outflow using four independent axioms. Besides the well-known axioms of anonymity and positive responsiveness we introduce outflow monotonicity – meaning that in pairwise comparison between two nodes, a node is not doing worse in case its own outflow does not decrease and the other node’s outflow does not increase – and order preservation – meaning that adding two weighted digraphs such that the pairwise ranking between two nodes is the same in both weighted digraphs, then this is also their pairwise ranking in the ‘sum’ weighted digraph. The outflow ranking method generalizes the ranking by outdegree for directed graphs, and therefore also generalizes the ranking by Copeland score for tournaments.     

完全对称有向图Dn的偶长圈分解

证明了将奇数阶完全对称有向图Dn分拆为偶长有向圈的必要条件也是充分的。     

Backbone colorings for graphs: Tree and path backbones

We introduce and study backbone colorings, a variation on classical vertex colorings: Given a graph G = (V,E) and a spanning subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex coloring V → {1,2,…} of G in which the colors assigned to adjacent vertices in H differ by at least two. We study the cases where the backbone is either a spanning tree or a spanning path. We show that for tree backbones of G the number of colors needed for a backbone coloring of G can roughly differ by a multiplicative factor of at most 2 from the chromatic number χ(G); for path backbones this factor is roughly . We show that the computational complexity of the problem “Given a graph G, a spanning tree T of G, and an integer ?, is there a backbone coloring for G and T with at most ? colors?” jumps from polynomial to NP‐complete between ? = 4 (easy for all spanning trees) and ? = 5 (difficult even for spanning paths). We finish the paper by discussing some open problems. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 137–152, 2007     

A linear-time algorithm for the longest path problem in rectangular grid graphs

The longest path problem is a well-known NP-hard problem and so far it has been solved polynomially only for a few classes of graphs. In this paper, we give a linear-time algorithm for finding a longest path between any two given vertices in a rectangular grid graph.     

A forbidden subgraph characterization of line-polar bipartite graphs

A graph is polar if the vertex set can be partitioned into A and B in such a way that the subgraph induced by A is a complete multipartite graph and the subgraph induced by B is a disjoint union of cliques. Polar graphs are a common generalization of bipartite, cobipartite, and split graphs. However, recognizing polar graphs is an NP-complete problem in general. This led to the study of the polarity of special classes of graphs such as cographs and chordal graphs, cf. Ekim et al. (2008) [7] and [5]. In this paper, we study the polarity of line graphs and call a graph line-polar if its line graph is polar. We characterize line-polar bipartite graphs in terms of forbidden subgraphs. This answers a question raised in the fist reference mentioned above. Our characterization has already been used to develop a linear time algorithm for recognizing line-polar bipartite graphs, cf. Ekim (submitted for publication) [6].