A forbidden induced subgraph characterization of distance-hereditary 5-leaf powers

A graph G is a k-leaf power if there is a tree T such that the vertices of G are the leaves of T and two vertices are adjacent in G if and only if their distance in T is at most k. In this situation T is called a k-leaf root of G. Motivated by the search for underlying phylogenetic trees, the notion of a k-leaf power was introduced and studied by Nishimura, Ragde and Thilikos and subsequently in various other papers. While the structure of 3- and 4-leaf powers is well understood, for k≥5 the characterization of k-leaf powers remains a challenging open problem.In the present paper, we give a forbidden induced subgraph characterization of distance-hereditary 5-leaf powers. Our result generalizes known characterization results on 3-leaf powers since these are distance-hereditary 5-leaf powers.     

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).     

Polynomial kernels for 3-leaf power graph modification problems

A graph G=(V,E) is a 3-leaf power iff there exists a tree T the leaf set of which is V and such that uv∈E iff u and v are at distance at most 3 in T. The 3-leaf power graph edge modification problems, i.e. edition (also known as the closest 3-leaf power), completion and edge-deletion are FPT when parameterized by the size of the edge set modification. However, polynomial kernels were known for none of these three problems. For each of them, we provide kernels with O(k³) vertices that can be computed in linear time. We thereby answer an open problem first mentioned by Dom et al. (2004) [8].     

Boxicity of Leaf Powers

The boxicity of a graph G, denoted as boxi(G), is defined as the minimum integer t such that G is an intersection graph of axis-parallel t-dimensional boxes. A graph G is a k-leaf power if there exists a tree T such that the leaves of the tree correspond to the vertices of G and two vertices in G are adjacent if and only if their corresponding leaves in T are at a distance of at most k. Leaf powers are used in the construction of phylogenetic trees in evolutionary biology and have been studied in many recent papers. We show that for a k-leaf power G, boxi(G)??? k?1. We also show the tightness of this bound by constructing a k-leaf power with boxicity equal to k?1. This result implies that there exist strongly chordal graphs with arbitrarily high boxicity which is somewhat counterintuitive.     

Triangular line graphs and word sense disambiguation

Linguists often represent the relationships between words in a collection of text as an undirected graph G=(V,E), where V is the vocabulary and vertices are adjacent in G if and only if the words that they represent co-occur in a relevant pattern in the text. Ideally, the words with similar meanings give rise to the vertices of a component of the graph. However, many words have several distinct meanings, preventing components from characterizing distinct semantic fields. This paper examines how the structural properties of triangular line graphs motivate the use of a clustering coefficient on the triangular line graph, thereby helping to identify polysemous words. The triangular line graph of G, denoted by T(G), is the subgraph of the line graph of G where two vertices are adjacent if the corresponding edges in G belong to a K₃.     

Total restrained domination numbers of trees

For a given connected graph G=(V,E), a set D_tr⊆V(G) is a total restrained dominating set if it is dominating and both 〈D_tr〉 and 〈V(G)-D_tr〉 do not contain isolate vertices. The cardinality of the minimum total restrained dominating set in G is the total restrained domination number and is denoted by γ_tr(G). In this paper we characterize the trees with equal total and total restrained dominating numbers and give a lower bound on the total restrained dominating number of a tree T in terms of its order and the number of leaves of T.     

New upper bounds on the spectral radius of unicyclic graphs

Let G=(V(G),E(G)) be a unicyclic simple undirected graph with largest vertex degree Δ. Let C_r be the unique cycle of G. The graph G-E(C_r) is a forest of r rooted trees T₁,T₂,…,T_r with root vertices v₁,v₂,…,v_r, respectively. Let

     

Can transitive orientation make sandwich problems easier?

A graph G_s=(V,E_s) is a sandwich for a pair of graphs G_t=(V,E_t) and G=(V,E) if E_t⊆E_s⊆E. A sandwich problem asks for the existence of a sandwich graph having an expected property. In a seminal paper, Golumbic et al. [Graph sandwich problems, J. Algorithms 19 (1995) 449-473] present many results on sub-families of perfect graphs. We are especially interested in comparability (resp., co-comparability) graphs because these graphs (resp., their complements) admit one or more transitive orientations (each orientation is a partially ordered set or poset). Thus, fixing the orientations of the edges of G_t and G restricts the number of possible sandwiches. We study whether adding an orientation can decrease the complexity of the problem. Two different types of problems should be considered depending on the transitivity of the orientation: the poset sandwich problems and the directed sandwich problems. The orientations added to both graphs G and G_s are transitive in the first type of problem but arbitrary for the second type.     

Maximum nullity of outerplanar graphs and the path cover number

Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[a_ij] with a_ij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).     

Some bounds on the p-domination number in trees

Let p be a positive integer and G=(V,E) a graph. A subset S of V is a p-dominating set if every vertex of V-S is dominated at least p times, and S is a p-dependent set of G if the subgraph induced by the vertices of S has maximum degree at most p-1. The minimum cardinality of a p-dominating set a of G is the p-domination number γ_p(G) and the maximum cardinality of a p-dependent set of G is the p-dependence number β_p(G). For every positive integer p?2, we show that for a bipartite graph G, γ_p(G) is bounded above by (|V|+|Y_p|)/2, where Y_p is the set of vertices of G of degree at most p-1, and for every tree T, γ_p(T) is bounded below by β_p-1(T). Moreover, we characterize the trees achieving equality in each bound.     

On the spectra of certain rooted trees

Let T be an unweighted tree with vertex root v which is the union of two trees T₁=(V₁,E₁), T₂=(V₂,E₂) such that V₁ ∩ V₂ = {v} and T₁ and T₂ have the property that the vertices in each of their levels have equal degree. We characterize completely the eigenvalues of the adjacency matrix and of the Laplacian matrix of T. They are the eigenvalues of symmetric tridiagonal matrices whose entries are given in terms of the vertex degrees. Moreover, we give some results about the multiplicity of the eigenvalues. Applications to some particular trees are developed.     

Planar unit-distance graphs having planar unit-distance complement

A graph G=(V,E) is called a unit-distance graph in the plane if there is an embedding of V into the plane such that every pair of adjacent vertices are at unit distance apart. If an embedding of V satisfies the condition that two vertices are adjacent if and only if they are at unit distance apart, then G is called a strict unit-distance graph in the plane. A graph G is a (strict) co-unit-distance graph, if both G and its complement are (strict) unit-distance graphs in the plane. We show by an exhaustive enumeration that there are exactly 69 co-unit-distance graphs (65 are strict co-unit-distance graphs), 55 of which are connected (51 are connected strict co-unit-distance graphs), and seven are self-complementary.     

A substitution theorem for graceful trees and its applications

A graceful labeling of a graph G=(V,E) assigns |V| distinct integers from the set {0,…,|E|} to the vertices of G so that the absolute values of their differences on the |E| edges of G constitute the set {1,…,|E|}. A graph is graceful if it admits a graceful labeling. The forty-year old Graceful Tree Conjecture, due to Ringel and Kotzig, states that every tree is graceful.We prove a Substitution Theorem for graceful trees, which enables the construction of a larger graceful tree through combining smaller and not necessarily identical graceful trees. We present applications of the Substitution Theorem, which generalize earlier constructions combining smaller trees.     

Locating and total dominating sets in trees

A set S of vertices in a graph G=(V,E) is a total dominating set of G if every vertex of V is adjacent to a vertex in S. We consider total dominating sets of minimum cardinality which have the additional property that distinct vertices of V are totally dominated by distinct subsets of the total dominating set.     

An efficient condition for a graph to be Hamiltonian

Let G=(V,E) be a 2-connected simple graph and let d_G(u,v) denote the distance between two vertices u,v in G. In this paper, it is proved: if the inequality d_G(u)+d_G(v)?|V(G)|-1 holds for each pair of vertices u and v with d_G(u,v)=2, then G is Hamiltonian, unless G belongs to an exceptional class of graphs. The latter class is described in this paper. Our result implies the theorem of Ore [Note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55]. However, it is not included in the theorem of Fan [New sufficient conditions for cycles in graph, J. Combin. Theory Ser. B 37 (1984) 221-227].     

A greedy approximation algorithm for the group Steiner problem

In the group Steiner problem we are given an edge-weighted graph G=(V,E,w) and m subsets of vertices . Each subset g_i is called a group and the vertices in ?_ig_i are called terminals. It is required to find a minimum weight tree that contains at least one terminal from every group.We present a poly-logarithmic ratio approximation for this problem when the input graph is a tree. Our algorithm is a recursive greedy algorithm adapted from the greedy algorithm for the directed Steiner tree problem [Approximating the weight of shallow Steiner trees, Discrete Appl. Math. 93 (1999) 265-285, Approximation algorithms for directed Steiner problems, J. Algorithms 33 (1999) 73-91]. This is in contrast to earlier algorithms that are based on rounding a linear programming based relaxation for the problem [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66-84, preliminary version in Proceedings of SODA, 1998 pp. 253-259, On directed Steiner trees, Proceedings of SODA, 2002, pp. 59-63]. We answer in positive a question posed in [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66-84, preliminary version in Proceedings of SODA, 1998 pp. 253-259] on whether there exist good approximation algorithms for the group Steiner problem that are not based on rounding linear programs. For every fixed constant ε>0, our algorithm gives an approximation in polynomial time. Approximation algorithms for trees can be extended to arbitrary undirected graphs by probabilistically approximating the graph by a tree. This results in an additional multiplicative factor of in the approximation ratio, where |V| is the number of vertices in the graph. The approximation ratio of our algorithm on trees is slightly worse than the ratio of O(log(max_i|g_i|)·logm) provided by the LP based approaches.     

Subdivided trees are integral sum graphs

A graph G=(V,E) is an integral sum graph (ISG) if there exists a labeling S(G)⊂Z such that V=S(G) and for every pair of distinct vertices u,v∈V, uv is an edge if and only if u+v∈V. A vertex in a graph is called a fork if its degree is not 2. In 1998, Chen proved that every tree whose forks are at distance at least 4 from each other is an ISG. In 2004, He et al. reduced the distance to 3. In this paper we reduce the distance further to 2, i.e. we prove that every tree whose forks are at least distance 2 apart is an ISG.     

A short and unified proof of Yu et al.?s two results on the eccentric distance sum

The eccentric distance sum (EDS) is a novel topological index that offers a vast potential for structure activity/property relationships. For a connected graph G, the eccentric distance sum is defined as ξd(G)=_∑v∈V(G)ecG(v)DG(v), where ecG(v) is the eccentricity of a vertex v in G and DG(v) is the sum of distances of all vertices in G from v. More recently, Yu et al. [G. Yu, L. Feng, A. Ili?, On the eccentric distance sum of trees and unicyclic graphs, J. Math. Anal. Appl. 375 (2011) 99-107] proved that for an n-vertex tree T, ξd(T)?4n²−9n+5, with equality holding if and only if T is the n-vertex star Sn, and for an n-vertex unicyclic graph G, ξd(G)?4n²−9n+1, with equality holding if and only if G is the graph obtained by adding an edge between two pendent vertices of n-vertex star. In this note, we give a short and unified proof of the above two results.     

Total domination subdivision numbers of trees

A set S of vertices in a graph G is a total dominating set of G if every vertex is adjacent to a vertex in S. The total domination number γ_t(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd_γt(G) of a graph G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the total domination number. Haynes et al. (J. Combin. Math. Combin. Comput. 44 (2003) 115) showed that for any tree T of order at least 3, 1?sd_γt(T)?3. In this paper, we give a constructive characterization of trees whose total domination subdivision number is 3.