Isometric Subgraphs Research Articles

Rearrangement operations on unrooted phylogenetic networks

Rearrangement operations transform a phylogenetic tree into another one and hence induce a metric on the space of phylogenetic trees. Popular operations for unrooted phylogenetic trees are NNI (nearest neighbour interchange), SPR (subtree prune and regraft), and TBR (tree bisection and reconnection). Recently, these operations have been extended to unrooted phylogenetic networks, which are generalisations of phylogenetic trees that can model reticulated evolutionary relationships. Here, we study global and local properties of spaces of phylogenetic networks under these three operations. In particular, we prove connectedness and asymptotic bounds on the diameters of spaces of different classes of phylogenetic networks, including tree-based and level-k networks. We also examine the behaviour of shortest TBR-sequence between two phylogenetic networks in a class, and whether the TBR-distance changes if intermediate networks from other classes are allowed: for example, the space of phylogenetic trees is an isometric subgraph of the space of phylogenetic networks under TBR. Lastly, we show that computing the TBR-distance and the PR-distance of two phylogenetic networks is NP-hard.

Obstructions to a small hyperbolicity in Helly graphs

The δ-hyperbolicity of a graph is defined by a simple 4-point condition: for any four vertices u, v, w, and x, the two larger of the distance sums d(u,v)+d(w,x), d(u,w)+d(v,x), and d(u,x)+d(v,w) differ by at most 2δ≥0. Hyperbolicity can be viewed as a measure of how close a graph is to a tree metrically; the smaller the hyperbolicity of a graph, the closer it is metrically to a tree. A graph G is Helly if its disks satisfy the Helly property, i.e., every family of pairwise intersecting disks in G has a common intersection. It is known that for every graph G there exists the smallest Helly graph H(G) into which G isometrically embeds (H(G) is called the injective hull of G) and the hyperbolicity of H(G) is equal to the hyperbolicity of G. Motivated by this, we investigate structural properties of Helly graphs that govern their hyperbolicity and identify three isometric subgraphs of the King-grid as structural obstructions to a small hyperbolicity in Helly graphs.

The self-concatenation of isometric strings is isometric

The generalized Fibonacci Qd(f) is defined as the graph obtained from the d-cube Qd by removing all vertices that contain a given binary string f as a contiguous substring. This idea was introduced by Ilić, Klavžar and Rho. A binary string f is called isometric if Qd(f) is an isometric subgraph of Qd for all d≥1, otherwise it is called non-isometric. In this paper, we prove that a string f is isometric if and only if fn is isometric for any n≥1. This result can help us to construct more isometric strings and significantly increase the number of isometric strings.

All good (bad) words consisting of 5 blocks

Generalized Fibonacci cube Qd(f), introduced by Ilic, Klavžar and Rho, is the graph obtained from the hypercube Qd by removing all vertices that contain f as factor. A word f is good if Qd(f) is an isometric subgraph of Qd for all d ≥ 1, and bad otherwise. A non-extendable sequence of contiguous equal digits in a word μ is called a block of μ. Ilic, Klavžar and Rho shown that all the words consisting of one block are good, and all the words consisting of three blocks are bad. So a natural problem is to study the words consisting of other odd number of blocks. In the present paper, a necessary condition for a word consisting of odd number of blocks being good is given, and all the good (bad) words consisting of 5 blocks is determined.

A negative answer to a problem on generalized Fibonacci cubes

Generalized Fibonacci cube Qn(f) is the graph obtained from the n-cube Qn by removing all vertices that contain a given binary string f as a consecutive substring. A binary string f is called bad if Qn(f) is not an isometric subgraph of Qn for some n, and the smallest such integer n, denoted by B(f), is called the index of f. Ilić, Klavžar and Rho posed a problem that if Qn(f) is not an isometric subgraph of Qn, is there a dimension n′ such that Qn(f) can be isometrically embedded into Qn′? We give a negative answer to this problem by showing that if f is bad, then for any n≥B(f), Qn(f) cannot be isometrically embedded to any hypercube.

Hamiltonian Laceability of Hypercubes Without Isometric Subgraphs

Locke (Am Math Mon 108:668, 2001) conjectured that the n-dimensional hypercube $$Q_n$$Qn with the set F of 2k removed vertices half from each bipartition set, where $$n \ge k + 2$$nźk+2 and $$k \ge 1$$kź1, is Hamiltonian. So far the conjecture remains open although partial results are known; some of them with additional conditions on the set F. We explore Hamiltonian properties of $$Q_n - F$$Qn-F if the set of faulty vertices F forms either an isometric cycle of $$Q_n$$Qn or an isometric tree of $$Q_n$$Qn. We study a more general problem. A bipartite graph G is Hamiltonian laceable if either (a) its bipartition sets are of equal size and for each pair of vertices x, y from different bipartition sets there exists a Hamiltonian path between x and y, or (b) its bipartition sets differ in sizes by one and for each pair of vertices x, y from the larger bipartition set there exists a Hamiltonian path between x and y. In particular, we show that if C is an isometric cycle in $$Q_n$$Qn for $$n \ge 5$$nź5, then $$Q_n - V(C)$$Qn-V(C) is Hamiltonian laceable. This allows us to remove up to 2n faulty vertices. Furthermore, if T is balanced isometric tree in $$Q_n$$Qn, then for $$n \ge 4$$nź4 the graph $$Q_n - V(T)$$Qn-V(T) is Hamiltonian laceable. Finally, if T is an almost-balanced isometric tree in $$Q_n$$Qn, then for $$n \ge 5$$nź5 the graph $$Q_n - V(T)$$Qn-V(T) is Hamiltonian laceable. Thus our results support Locke hypothesis.

The structures of bad words

The generalized Fibonacci cube Qd(f) is the graph obtained from the hypercube Qd by removing all vertices that contain a given binary word f. A word f is called good if Qd(f) is an isometric subgraph of Qd for all d≥1, and bad otherwise. The index B(f) of a word f is the smallest integer d such that Qd(f) is not an isometric subgraph of Qd. Klavžar and Shpectorov showed that a bad word has a 2-error overlap. In this paper, we show that if a word has a 2-error overlap then it is bad, and we characterize the structures of all bad words. Ilić, Klavžar and Rho showed that a word f is bad if there exist p-critical words for Qd(f) for some d. We find that a word is bad if and only if there exists only a pair of 2-critical words, or only a pair of 3-critical words, or both only a pair of 2-critical words and only a pair of 3-critical words for QB(f)(f). We also design an O(|f|3) algorithm which can output the index B(f) of f, and give all pairs of p-critical words for QB(f)(f).

To the theory of [formula omitted]-ary Steiner and other-type trades

We introduce the concept of a clique bitrade, which generalizes several known types of bitrades, including latin bitrades, Steiner T(k−1,k,v) bitrades, extended 1-perfect bitrades. For a distance-regular graph, we show a one-to-one correspondence between the clique bitrades that meet the weight-distribution lower bound on the cardinality and the bipartite isometric subgraphs that are distance-regular with certain parameters. As an application of the results, we find the minimum cardinality of q-ary Steiner Tq(k−1,k,v) bitrades and show a connection of minimum such bitrades with dual polar subgraphs of the Grassmann graph Jq(v,k).

Proofs of two conjectures on generalized Fibonacci cubes

A binary string f is a factor of string u if f appears as a sequence of |f| consecutive bits of u, where |f| denotes the length of f. Generalized Fibonacci cube Qd(f) is the graph obtained from the d-cube Qd by removing all vertices that contain a given binary string f as a factor. A binary string f is called good if Qd(f) is an isometric subgraph of Qd for all d≥1, it is called bad otherwise. The index of a binary string f, denoted by B(f), is the smallest integer d such that Qd(f) is not an isometric subgraph of Qd. Ilić, Klavžar and Rho conjectured that B(f)<2|f| for any bad string f. They also conjectured that if Qd(f) is an isometric subgraph of Qd, then Qd(ff) is an isometric subgraph of Qd. We confirm these two conjectures by obtaining a basic result: if there exist p-critical words for QB(f)(f), then p=2 or p=3.

Guarded subgraphs and the domination game

Graph Theory We introduce the concept of guarded subgraph of a graph, which as a condition lies between convex and 2-isometric subgraphs and is not comparable to isometric subgraphs. Some basic metric properties of guarded subgraphs are obtained, and then this concept is applied to the domination game. In this game two players, Dominator and Staller, alternate choosing vertices of a graph, one at a time, such that each chosen vertex enlarges the set of vertices dominated so far. The aim of Dominator is that the graph is dominated in as few steps as possible, while the aim of Staller is just the opposite. The game domination number is the number of vertices chosen when Dominator starts the game and both players play optimally. The main result of this paper is that the game domination number of a graph is not smaller than the game domination number of any guarded subgraph. Several applications of this result are presented.

Tree-like partial Hamming graphs

Tree-like partial cubes were introduced in [B. Bresar, W. Imrich, S. Klavžar, Tree-like isometric subgraphs of hypercubes, Discuss. Math. Graph Theory, 23 (2003), 227–240] as a generalization of median graphs. We present some incorrectnesses from that article. In particular we point to a gap in the proof of the theorem about the dismantlability of the cube graph of a tree-like partial cube and give a new proof of that result, which holds also for a bigger class of graphs, so called tree-like partial Hamming graphs. We investigate these graphs and show some results which imply previously-known results on tree-like partial cubes. For instance, we characterize tree-like partial Hamming graphs and prove that every tree-like partial Hamming graph G contains a Hamming graph that is invariant under every automorphism of G. The latter result is a direct consequence of the result about the dismantlability of the intersection graph of maximal Hamming graphs of a tree-like partial Hamming graph.

Generalized Lucas cubes

Let f be is a binary string and d?1. Then the generalized Lucas cube Qd(f?)is introduced as the graph obtained from the Qd by removing all vertices that have a circulation containing f as a substring. The question for which f and d, the generalized Lucas cube Qd(f?) is an isometric subgraph of the d-cube Qd is solved for all binary strings of length at most five. Several isometrically embeddable and non-embeddable infinite series where f is of arbitrary length are given. Some structural properties of generalized Lucas cubes are also presented.

On the rainbow connection of Cartesian products and their subgraphs

Rainbow connection number of Cartesian products and their subgraphs are considered. Previously known bounds are compared and non-existence of such bounds for subgraphs of products are discussed. It is shown that the rainbow connection number of an isometric subgraph of a hypercube is bounded above by the rainbow connection number of the hypercube. Isometric subgraphs of hypercubes with the rainbow connection number as small as possible compared to the rainbow connection of the hypercube are constructed. The concept of c-strong rainbow connected coloring is introduced. In particular, it is proved that the so-called �-coloring of an isometric subgraph of a hypercube is its unique optimal c-strong rainbow connected coloring.

Weak geodesic topology and fixed finite subgraph theorems in infinite partial cubes II. Fixed subgraph properties and infinite treelike partial cubes

We prove that if a partial cube G contains no divergent rays, then: (i) there exists a non-empty finite convex subgraph of G which is fixed by every automorphism of G; (ii) every self-contraction of G fixes a non-empty finite isometric subgraph of G; (iii) for every commuting family F of self-contractions of G, there exists a non-empty finite isometric subgraph of G which is fixed by every element of F. Moreover, we show that those fixed subgraphs can be specified for some particular partial cubes, as is the case for median graphs and more generally for netlike partial cubes. We give new examples of such special graphs by extending the finite concept of treelike partial cubes [B. Brešar et al., Tree-like isometric subgraphs of hypercubes, Discussiones Mathematicae–Graph Theory 23 (2003), 227–240] to the infinite case, and by introducing what we call faithful-treelike, convex-treelike and strongly treelike partial cubes.

Generalized Fibonacci cubes

Generalized Fibonacci cube Qd(f) is introduced as the graph obtained from the d-cube Qd by removing all vertices that contain a given binary string f as a substring. In this notation, the Fibonacci cube Γd is Qd(11). The question whether Qd(f) is an isometric subgraph of Qd is studied. Embeddable and non-embeddable infinite series are given. The question is completely solved for strings f of length at most five and for strings consisting of at most three blocks. Several properties of the generalized Fibonacci cubes are deduced. Fibonacci cubes are, besides the trivial cases Qd(10) and Qd(01), the only generalized Fibonacci cubes that are median closed subgraphs of the corresponding hypercubes. For admissible strings f, the f-dimension of a graph is introduced. Several problems and conjectures are also listed.

Hyperbolicity and Chordality of a Graph

Let $G$ be a connected graph with the usual shortest-path metric $d$. The graph $G$ is $\delta$-hyperbolic provided for any vertices $x,y,u,v$ in it, the two larger of the three sums $d(u,v)+d(x,y),d(u,x)+d(v,y)$ and $d(u,y)+d(v,x)$ differ by at most $2\delta.$ The graph $G$ is $k$-chordal provided it has no induced cycle of length greater than $k.$ Brinkmann, Koolen and Moulton find that every $3$-chordal graph is $1$-hyperbolic and that graph is not $\frac{1}{2}$-hyperbolic if and only if it contains one of two special graphs as an isometric subgraph. For every $k\geq 4,$ we show that a $k$-chordal graph must be $\frac{\lfloor\frac{k}{2}\rfloor}{2}$-hyperbolic and there does exist a $k$-chordal graph which is not $\frac{\lfloor \frac{k-2}{2}\rfloor}{2}$-hyperbolic. Moreover, we prove that a $5$-chordal graph is $\frac{1}{2}$-hyperbolic if and only if it does not contain any of a list of five special graphs as an isometric subgraph.

Blocks and Cut Vertices of the Buneman Graph

Given a set $\Sigma$ of bipartitions of some finite set X of cardinality at least 2, one can associate to $\Sigma$ a canonical X-labeled graph $\mathcal{B}(\Sigma)$, called the Buneman graph. This graph has several interesting mathematical properties—for example, it is a median network and therefore an isometric subgraph of a hypercube. It is commonly used as a tool in studies of DNA sequences gathered from populations. In this paper, we present some results concerning the cut vertices of $\mathcal{B}(\Sigma)$, i.e., vertices whose removal disconnect the graph, as well as its blocks or 2-connected components—results that yield, in particular, an intriguing generalization of the well-known fact that $\mathcal{B}(\Sigma)$ is a tree if and only if any two splits in $\Sigma$ are compatible.

Netlike partial cubes, V: Completion and netlike classes

We define a completion of a netlike partial cube G by replacing each convex 2 n -cycle C of G with n ≥ 3 by an n -cube admitting C as an isometric cycle. We prove that a completion of G is a median graph if and only if G has the Median Cycle Property (MCP) (see N. Polat, Netlike partial cubes III. The Median Cycle Property, Discrete Math.). In fact any completion of a netlike partial cube having the MCP is defined by a universal property and turns out to be a minimal median graph containing G as an isometric subgraph. We show that the completions of the netlike partial cubes having the MCP preserves the principal constructions of these graphs, such as: netlike subgraphs, gated amalgams and expansions. Conversely any netlike partial cube having the MCP can be obtained from a median graph by deleting some particular maximal finite hypercubes. We also show that, given a netlike partial cube G having the MCP, the class of all netlike partial cubes having the MCP whose completions are isomorphic to those of G share different properties, such as: depth, lattice dimension, semicube graph and crossing graph.

Connected graphs as subgraphs of Cayley graphs: Conditions on Hamiltonicity

Let Γ be a connected simple graph, let V ( Γ ) and E ( Γ ) denote the vertex-set and the edge-set of Γ , respectively, and let n = | V ( Γ ) | . For 1 ≤ i ≤ n , let e i be the element of elementary abelian group Z 2 n which has 1 in the i th coordinate, and 0 in all other coordinates. Assume that V ( Γ ) = { e i ∣ 1 ≤ i ≤ n } . We define a set Ω by Ω = { e i + e j ∣ { e i , e j } ∈ E ( Γ ) } , and let Cay Γ denote the Cayley graph over Z 2 n with respect to Ω . It turns out that Cay Γ contains Γ as an isometric subgraph. In this paper, the relations between the spectra of Γ and Cay Γ are discussed. Some conditions on the existence of Hamilton paths and cycles in Γ are obtained.

Isometric embeddings of subdivided connected graphs into hypercubes

Isometric subgraphs of hypercubes are known as partial cubes. These graphs have first been investigated by Graham and Pollack [R.L. Graham, H. Pollack, On the addressing problem for loop switching, Bell System Technol. J. 50 (1971) 2495–2519; and D. Djoković, Distance preserving subgraphs of hypercubes, J. Combin. Theory Ser. B 14 (1973) 263–267]. Several papers followed with various characterizations of partial cubes. In this paper, we determine all subdivisions of a given configuration which can be embedded isometrically in the hypercube. More specifically, we deal with the case where this configuration is a connected graph of order 4, a complete graph of order 5 and the case of a k -fan F k ( k ≥ 3 ) .