Discrete Applied Mathematics最新文献

A graph $G=(V\left(G\right),E\left(G\right))$ is called a $k$-dot product graph if there is a function $f:V\left(G\right)⟶R^k$ such that for any two distinct vertices $u$ and $v$, one has $f\left(u\right).f\left(v\right)\ge 1$ if and only if $uv\in E\left(G\right)$. The minimum value $k$ such that $G$ is a $k$-dot product graph, is called the dot product dimension $\rho \left(G\right)$ of $G$. These concepts were introduced for the first time by Fiduccia, Scheinerman, Trenk and Zito. In this paper, we determine the dot product dimension of unicyclic graphs.

A $P_{\ge k}$-factor is a spanning subgraph $H$ of $G$ whose components are paths of order at least $k$. A graph $G$ is $P_{\ge k}$-factor uniform if for arbitrary $e_1,e_2\in E\left(G\right)$ with $e_1\ne e_2$, $G$ has a $P_{\ge k}$-factor containing $e_1$ and avoiding $e_2$. Liu first put forward the concept of $(P_{\ge k},n)$-critical uniform graph, that is, a graph $G$ is called $(P_{\ge k},n)$-critical uniform if the graph $G-V^{\prime }$ is $P_{\ge k}$-factor uniform for any $V^{\prime }\subseteq V\left(G\right)$ with $\left|V^{\prime }\right|=n$. In this paper, two new results on $(P_{\ge k},n)$-critical uniform graphs $(k=2,3)$ in terms of independence number and minimum degree are presented. Furthermore, we show the sharpness of the main results in this paper by structuring special counterexamples.

For an integer $k\ge 2$, a $k$-community structure in an undirected graph is a partition of its vertex set into $k$ sets called communities, each of size at least two, such that every vertex of the graph has proportionally at least as many neighbours in its own community as in any other community. In this paper, we give a necessary and sufficient condition for a forest on $n$ vertices to admit a $k$-community structure. Furthermore, we provide an $O\left(k^2\cdot n^2\right)$-time algorithm that computes such a $k$-community structure in a forest, if it exists. These results extend a result of Bazgan et al., 2018. We also show that if communities are allowed to have size one, then every forest with $n\ge k\ge 2$ vertices admits a $k$-community structure that can be found in time $O\left(k^2\cdot n^2\right)$. We then consider threshold graphs and show that every connected threshold graph admits a 2-community structure if and only if it is not isomorphic to a star; also if such a 2-community structure exists, we explain how to obtain it in linear time. We further describe an infinite family of disconnected threshold graphs, containing exactly one isolated vertex, that do not admit any 2-community structure. Finally, we present a new infinite family of connected graphs that may contain an even or an odd number of vertices without 2-community structures, even if communities are allowed to have size one.

Let $G$ be a nonseparable graph of order at least 3, in this paper, we prove that: (1) if every ear of an ear decomposition of $G$ is a path of length at least two, then $G$ is 3-colorable; and (2) if $H$ is the spanning subgraph of $G$ whose edges are those edges in the ears of length 1, then the chromatic number of $G$ can be bounded in terms of a parameter of $H$.

Our first result implies that there is a polynomial time test to ensure that a given nonseparable graph is 3-colorable. Our second result gives us the possibility to iterate the test with the nonseparable components of $H$.

A connected matching in a graph $G$ consists of a set of pairwise disjoint edges whose covered vertices induce a connected subgraph of $G$. While finding a connected matching of maximum cardinality is a well-solved problem, it is NP-hard to determine an optimal connected matching in an edge-weighted graph, even in the planar bipartite case. We present two mixed integer programming formulations and a sophisticated branch-and-cut scheme to find weighted connected matchings in general graphs. The formulations explore different polyhedra associated to this problem, including strong valid inequalities both from the matching polytope and from the connected subgraph polytope. We conjecture that one attains a tight approximation of the convex hull of connected matchings using our strongest formulation, and report encouraging computational results over DIMACS Implementation Challenge benchmark instances. The source code of the complete implementation is also made available.

The structure of minimal weight rainbow domination functions of cubic graphs are studied. Based on general observations for cubic graphs, generalized Petersen graphs $P(ck,k)$ are characterized whose 4- and 5-rainbow domination numbers equal the general lower bounds. As $t$-rainbow domination of cubic graphs for $t\ge 6$ is trivial, characterizations of such generalized Petersen graphs $P(ck,k)$ are known for all $t$-rainbow domination numbers. In addition, new upper bounds for 4- and 5-rainbow domination numbers that are valid for all $P(ck,k)$ are provided.

An orientation of a graph is semi-transitive if it is acyclic and shortcut-free. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs generalise several important classes of graphs and they are precisely the class of word-representable graphs studied extensively in the literature.

The Mycielski graph of an undirected graph is a larger graph, constructed in a certain way, that maintains the property of being triangle-free but enlarges the chromatic number. These graphs are important as they allow to prove the existence of triangle-free graphs with arbitrarily large chromatic number. An extended Mycielski graph is a certain natural extension of the notion of a Mycielski graph that we introduce in this paper.

In this paper we characterise completely semi-transitive extended Mycielski graphs and Mycielski graphs of comparability graphs. We also conjecture a complete characterisation of semi-transitive Mycielski graphs. Our studies are a far-reaching extension of the result of Kitaev and Pyatkin on non-semi-transitive orientability of the Mycielski graph $\mu \left(C_5\right)$ of the cycle graph $C_5$. Using a recent result of Kitaev and Sun, we shorten the length of the original proof of non-semi-transitive orientability of $\mu \left(C_5\right)$ from 2 pages to a few lines.

The total graph $T\left(G\right)$ of a graph $G$ has vertex set $V\left(T\left(G\right)\right)=V\left(G\right)\cup E\left(G\right)$, and two vertices in $T\left(G\right)$ are adjacent if and only if their corresponding elements are either adjacent or incident in $G$. The total graph operation can be used to generate large dense graphs (networks). In this paper, some spectral properties of total graphs are studied. We give expressions for the number of eigenvalues of $T\left(G\right)$ belong to the interval $(-2,\infty )$ and $(-\infty ,-2)$, and use the expressions to derive a lower bound on the clique partition number of $T\left(G\right)$. Expressions for the multiplicity and the eigenspace of eigenvalue $-2$ of $T\left(G\right)$ are obtained. We also give a formula for the characteristic polynomial of $T\left(G\right)$ in terms of the adjacency matrix and signless Laplacian matrix of $G$, and derive some properties for the Perron vector of $T\left(G\right)$.

Phylogenetic networks play an important role in evolutionary biology as, other than phylogenetic trees, they can be used to accommodate reticulate evolutionary events such as horizontal gene transfer and hybridization. Recent research has provided a lot of progress concerning the reconstruction of such networks from data as well as insight into their graph theoretical properties. However, methods and tools to quantify structural properties of networks or differences between them are still very limited. For example, for phylogenetic trees, it is common to use balance indices to draw conclusions concerning the underlying evolutionary model, and more than twenty such indices have been proposed and are used for different purposes. One of the most frequently used balance index for trees is the so-called total cophenetic index, which has several mathematically and biologically desirable properties. For networks, on the other hand, balance indices are to-date still scarce.

In this contribution, we introduce the weighted total cophenetic index as a generalization of the total cophenetic index for trees to make it applicable to general phylogenetic networks. As we shall see, this index can be determined efficiently and behaves in a mathematical sound way, i.e., it satisfies so-called locality and recursiveness conditions. In addition, we analyze its extremal properties and, in particular, we investigate its maxima and minima as well as the structure of networks that achieve these values within the space of so-called level-1 networks. We finally briefly compare this novel index to the two other network balance indices available so-far.

The median function is a location/consensus function that maps any profile $\pi$ (a finite multiset of vertices) to the set of vertices that minimize the distance sum to vertices from $\pi$. The median function satisfies several simple axioms: Anonymity (A), Betweeness (B), and Consistency (C). McMorris, Mulder, Novick and Powers (2015) defined the ABC-problem for consensus functions on graphs as the problem of characterizing the graphs (called, ABC-graphs) for which the unique consensus function satisfying the axioms (A), (B), and (C) is the median function. In this paper, we show that modular graphs with $G^2$-connected medians (in particular, bipartite Helly graphs) are ABC-graphs. On the other hand, the addition of some simple local axioms satisfied by the median function in all graphs (axioms (T), and (T${}_2$)) enables us to show that all graphs with connected median (comprising Helly graphs, median graphs, basis graphs of matroids and even $\Delta$-matroids) are ABCT-graphs and that benzenoid graphs are ABCT${}_2$-graphs. McMorris et al (2015) proved that the graphs satisfying the pairing property (called the intersecting-interval property in their paper) are ABC-graphs. We prove that graphs with the pairing property constitute a proper subclass of bipartite Helly graphs and we discuss the complexity status of the recognition problem of such graphs.