An (m,n)-colored mixed graph G is a graph with its arcs having one of the m different colors and edges having one of the n different colors. A homomorphism f of an (m,n)colored mixed graph G to an (m,n)-colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f(u)f(v) is an arc (edge) of color c in H . The (m,n)-colored mixed chromatic number χ(m,n)(G) of...

We consider acyclic directed mixed graphs, in which directed edges (x → y) and bi-directed edges (x ↔ y) may occur. A simple extension of Pearl’s d-separation criterion, called m-separation, is applied to these graphs. We introduce a local Markov property which is equivalent to the global property resulting from the m-separation criterion.

Search games in graphs have attracted significant attention in recent years, and they have applications in securing computer networks against viruses and intruders. Since graph searching is an NP-hard problem, polynomial-time algorithms have been given for solving it on various graph classes. Most of these algorithms concern computing the node search number of a graph, and only few such algorit...

We consider orientation problems on mixed graphs in which the goal is to obtain a directed graph satisfying certain connectivity requirements.

Let A be the edge-node incidence matrix of a bipartite graph G = (U, V ;E), I be a subset the nodes of G, and b be a vector such that 2b is integral. We consider the following mixed-integer set: X(G, b, I) = {x : Ax ≥ b, x ≥ 0, xi integer for all i ∈ I}. We characterize conv(X(G, b, I)) in its original space. That is, we describe a matrix (A′, b′) such that conv(X(G, b, I)) = {x : A′x ≥ b′}. Th...

A k-stack (respectively, k-queue) layout of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing (non-nested) edges with respect to the vertex ordering. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed 1-stack 1-queue layout in which every edge is assigned to a stack or to a queue that use a common vertex orde...

A mixed graph is a graph with directed edges, called arcs, and undirected edges. A k-coloring of the vertices is proper if colors from {1, 2, . . . , k} are assigned to each vertex such that u and v have different colors if uv is an edge, and the color of u is less than or equal to (resp. strictly less than) the color of v if uv is an arc. The weak (resp. strong) chromatic polynomial of a mixed...

Graph orientation is a fundamental problem in graph theory that has recently arisen in the study of signaling-regulatory pathways in protein networks. Given a graph and a list of ordered source-target vertex pairs, it calls for assigning directions to the edges of the graph so as to maximize the number of pairs that admit a directed source-to-target path. When the input graph is undirected, a s...

In this paper we provide a principled approach to solve a transductive classification problem involving a similar graph (edges tend to connect nodes with same labels) and a dissimilar graph (edges tend to connect nodes with opposing labels). Most of the existing methods, e.g., Information Regularization (IR), Weighted vote Relational Neighbor classifier (WvRN) etc, assume that the given graph i...