The problem of finding, in an edge-weighted bidirected graph $G=(V,E)$, a cycle with minimum mean weight of its edges generalizes similar problems for both directed and undirected graphs. (The problem is considered in two variants: for the cycles without repeated edges and for the cycles without repeated nodes.) In this note we develop an algorithm to solve this problem in $O(V^2 \min(V^2, E\log V))$-time (to compare: the complexity of an improved version of Barahona's algorithm for undirected cycles is $O(V^4)$). Our algorithm is based on a certain general approach to minimum mean problems and uses, as a subroutine, Gabow's algorithm for the minimum weight 2-factor problem in a graph. The problem admits a reformulation in terms of regular cycles in a skew-symmetric graph.
\emph{Bidirected graphs} (a sort of nonstandard graphs introduced by Edmonds and Johnson) provide a natural generalization to the notions of directed and undirected graphs. By a \emph{weakly (node- or edge-) acyclic} bidirected graph we mean such graph having no (node- or edge-) simple cycles. We call a bidirected graph \emph{strongly acyclic} if it has no cycles (even non-simple). Unlike the case of standard graphs, a bidirected graph may be weakly acyclic but still have non-simple cycles. Testing a given bidirected graph for weak acyclicity is a challenging combinatorial problem, which also …
Bidirected graphs (earlier studied by Edmonds, Johnson and, in equivalent terms of skew-symmetric graphs, by Tutte, Goldberg, Karzanov, and others) proved to be a useful unifying language for describing both flow and matching problems. In this paper we extend the notion of ear decomposition to the class of strongly connected bidirected graphs. In particular, our results imply Two Ear Theorem on matching covered graphs of Lovász and Plummer. The proofs given here are self-contained except for standard Barrier Theorem on skew-symmetric graphs.
We use $K^*_n$ to denote the bidirected complete graph on $n$ vertices. A nomadic Hamiltonian decomposition of $K^*_n$ is a Hamiltonian decomposition, with the additional property that ``nomads'' walk along the Hamiltonian cycles (moving one vertex per time step) without colliding. A nomadic near-Hamiltonian decomposition is defined similarly, except that the cycles in the decomposition have length $n-1$, rather than length $n$. J.A. Bondy asked whether these decompositions of $K^*_n$ exist for all $n$. We show that $K^*_n$ admits a nomadic near-Hamiltonian decomposition when $n\not\equiv 2\bm…
An edge uv in a graph Γ is directionally 2-signed (or, (2,d)-signed) by an ordered pair (a,b), a,b in {+,-}, if the label l(uv) = (a,b) from u to v, and l(vu) = (b,a) from v to u. Directionally 2-signed graphs are equivalent to bidirected graphs, where each end of an edge has a sign. A bidirected graph implies a signed graph, where each edge has a sign. We extend a theorem of Sriraj and Sampathkumar by proving that the signed graph is antibalanced (all even cycles and only even cycles have positive edge sign product) if, and only if, in the bidirected graph, after suitable reorientation of edg…
In a bidirected graph an edge has a direction at each end, so bidirected graphs generalize directed graphs. We generalize the definitions of transitive closure and transitive reduction from directed graphs to bidirected graphs by introducing new notions of bipath and bicircuit that generalize directed paths and cycles. We show how transitive reduction is related to transitive closure and to the matroids of the signed graph corresponding to the bidirected graph.
Let $G$ be a simple graph with no even cycle, called an odd-cycle graph. Cavers et al. [Cavers et al. Skew-adjacency matrices of graphs, Linear Algebra Appl. 436(2012), 4512--1829] showed that the spectral radius of $G^σ$ is the same for every orientation $σ$ of $G$, and equals the maximum matching root of $G$. They proposed a conjecture that the graphs which attain the maximum skew spectral radius among the odd-cycle graphs $G$ of order $n$ are isomorphic to the odd-cycle graph with one vertex degree $n-1$ and size $m=\lfloor 3(n-1)/2\rfloor$. This paper, by using the Kelmans transformation, …
Let $G=(V,E)$ be a finite undirected graph. Orient the edges of $G$ in an arbitrary way. A $2$-cycle on $G$ is a function $d : E^2\to \mathbb{Z}$ such for each edge $e$, $d(e, \cdot)$ and $d(\cdot, e)$ are circulations on $G$, and $d(e, f) = 0$ whenever $e$ and $f$ have a common vertex. We show that each $2$-cycle is a sum of three special types of $2$-cycles: cycle-pair $2$-cycles, Kuratowski $2$-cycles, and quad $2$-cycles. In case that the graph is Kuratowski connected, we show that each $2$-cycle is a sum of cycle-pair $2$-cycles and at most one Kuratowski $2$-cycle. Furthermore, if $G$ is…
Consider a simple graph in which a random walk begins at a given vertex. It moves at each step with equal probability to any neighbor of its current vertex, and ends when it has visited every vertex. We call such a random walk a random cover tour. It is well known that cycles and complete graphs have the property that a random cover tour starting at any vertex is equally likely to end at any other vertex. Ronald Graham asked whether there are any other graphs with this property. In 1993, Lászlo Lovász and Peter Winkler showed that cycles and complete graphs are the only undirected graphs with …
Let $G^σ$ be an oriented graph and $S(G^σ)$ be its skew-adjacency matrix, where $G$ is called the underlying graph of $G^σ$. The skew-rank of $G^σ$, denoted by $sr(G^σ)$, is the rank of $S(G^σ)$. Denote by $d(G)=|E(G)|-|V(G)|+θ(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $θ(G)$ are the edge number, vertex number and the number of connected components of $G$, respectively. Recently, Wong, Ma and Tian [European J. Combin. 54 (2016) 76--86] proved that $sr(G^σ)\leq r(G)+2d(G)$ for an oriented graph $G^σ$, where $r(G)$ is the rank of the adjacency matrix of $G$, and char…
Zero forcing is a process that colors the vertices of a graph blue by starting with some vertices blue and applying a color change rule. Throttling minimizes the sum of the number of initial blue vertices and the time to color the graph. In this paper, we study throttling for skew zero forcing. We characterize the graphs of order $n$ with skew throttling numbers $1, 2, n-1$, and $n$. We find the exact skew throttling numbers of paths, cycles, and balanced spiders with short legs. In addition, we find a sharp lower bound on skew throttling numbers in terms of the diameter.
Let $S(G^σ)$ be the skew-adjacency matrix of the oriented graph $G^σ$, which is obtained from a simple undirected graph $G$ by assigning an orientation $σ$ to each of its edges. The skew energy of an oriented graph $G^σ$ is defined as the sum of absolute values of all eigenvalues of $S(G^σ)$. For any positive integer $d$ with $3\leq d\leq n-3$, we determine the graph with minimal skew energy among all oriented bicyclic graphs that contain no vertex disjoint odd cycle of lengths $s$ and $l$ with $s+l\equiv 2(mod 4)$ on $n$ vertices with a given diameter $d$.
Let $G^σ$ be an orientation of a simple graph $G$. In this paper, the permanental polynomial of an oriented graph $G^σ$ is introduced. The coefficients of the permanental polynomial of $G^σ$ are interpreted in terms of the graph structure of $G^σ$, and it is proved that all orientations $G^σ$ of $G$ have the same permanental polynomial if and only if $G$ has no even cycles. Furthermore, the roots of the permanental polynomial of $G^σ$ are studied.
The minimum skew rank $mr^{-}(\mathbb{F},G)$ of a graph $G$ over a field $\mathbb{F}$ is the smallest possible rank among all skew symmetric matrices over $\mathbb{F}$, whose ($i$,$j$)-entry (for $i\neq j$) is nonzero whenever $ij$ is an edge in $G$ and is zero otherwise. We give some new properties of the minimum skew rank of a graph, including a characterization of the graphs $G$ with cut vertices over the infinite field $\mathbb{F}$ such that $mr^{-}(\mathbb{F},G)=4$, determination of the minimum skew rank of $k$-paths over a field $\mathbb{F}$, and an extending of an existing result to sho…
An oriented graph $G^σ$ is a digraph without loops or multiple arcs whose underlying graph is $G$. Let $S\left(G^σ\right)$ be the skew-adjacency matrix of $G^σ$ and $α(G)$ be the independence number of $G$. The rank of $S(G^σ)$ is called the skew-rank of $G^σ$, denoted by $sr(G^σ)$. Wong et al. [European J. Combin. 54 (2016) 76-86] studied the relationship between the skew-rank of an oriented graph and the rank of its underlying graph. In this paper, the correlation involving the skew-rank, the independence number, and some other parameters are considered. First we show that $sr(G^σ)+2α(G)\geq…
A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no odd hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge graphs saying that every Berge graph either is in a well understood basic class, or has some kind of decomposition. Then, Chudnovsky proved stronger theorems. One of them restricts the allowed decompositions to 2-joins and balanced skew partitions. We prove that the problem of deciding whether a graph has a balanced skew partition is NP-hard. We give an $O(n^9)$-…
To a directed graph without loops and 2-cycles, we can associate a skew-symmetric matrix with integer entries. Mutations of such skew-symmetric matrices, and more generally skew-symmetrizable matrices, have been defined in the context of cluster algebras by Fomin and Zelevinsky. The mutation class of a graph G is the set of all isomorphism classes of graphs that can be obtained from G by a sequence of mutations. A graph is called mutation-finite if its mutation class is finite. Fomin, Shapiro and Thurston constructed mutation-finite graphs from triangulations of oriented bordered surfaces with…
We study edge-labelings of the complete bidirected graph $\overset{\tiny\leftrightarrow}{K}_n$ with functions from the set $[d] = \{1, \dots, d\}$ to itself. We call a cycle in $\overset{\tiny\leftrightarrow}{K}_n$ a fixed-point cycle if composing the labels of its edges results in a map that has a fixed point, and we say that a labeling is fixed-point-free if no fixed-point cycle exists. For a given $d$, we ask for the largest value of $n$, denoted $R_f(d)$, for which there exists a fixed-point-free labeling of $\overset{\tiny\leftrightarrow}{K}_n$. Determining $R_f(d)$ for all $d >0$ is a na…
For all $n > k \ge 1$, we give formulas for the nullity $N(n,k)$ of the $n \times n$ skew-symmetric Toeplitz band matrix whose first $k$ superdiagonals have all entries $1$ and whose remaining superdiagonals have all entries $0$. This is accomplished by counting the number of cycles in certain directed graphs. As an application, for each fixed integer $z\ge 0$ and large fixed $k$, we give an asymptotic formula for the percentage of $n > k$ satisfying $N(n,k)=z$. For the purpose of rapid computation, an algorithm is devised that quickly computes $N(n,k)$ even for extremely large values of $n$ a…
The permanental polynomial of a graph $G$ is $π(G,x)\triangleq\mathrm{per}(xI-A(G))$. From the result that a bipartite graph $G$ admits an orientation $G^e$ such that every cycle is oddly oriented if and only if it contains no even subdivision of $K_{2,3}$, Yan and Zhang showed that the permanental polynomial of such a bipartite graph $G$ can be expressed as the characteristic polynomial of the skew adjacency matrix $A(G^e)$. In this paper we first prove that this equality holds only if the bipartite graph $G$ contains no even subdivision of $K_{2,3}$. Then we prove that such bipartite graphs …
A \emph{self-complementary} graph is a graph isomorphic to its complement. An isomorphism between $G$ and its complement, viewed as a permutation of $V(G)$, is then called an \emph{antimorphism}. A \emph{skew partition} of $G$ is a partition of $V(G)$ into 4 sets $A,B,C,D$ such that there is no edge between $A,B$ and every possible edge between $C,D$. A \emph{symmetric partition} of $G$ is a partition of $V(G)$ into 4 sets $A,B,C,D$ such that there is no edge between $A, D$, no edge between $B, C$, every possible edge between $A,B$ and every possible edge between $C,D$. We give a new proof of …
