Finite games in normal form and their mixed extensions are a corner stone of noncooperative game theory. Often  generic finite games and their mixed extensions are considered. But the properties which one expects in generic games and the existence of games with these properties are often treated only in passing. The paper considers strong properties and proves that generic games have these properties. The space of mixed strategy combinations is embedded in a natural way into a product of real projective spaces. All relevant hypersurfaces extend to this bigger space. The paper shows that for all games in the complement of a semialgebraic subset of codimension at least one all relevant hypersurfaces in the bigger space are smooth and maximally transversal. The proof uses the theorem of Sard and follows an argument of Khovanskii.

A proper \(k\)-coloring \(\alpha\) of a graph \(G\) induces a partition \(\Pi = \{C_1, C_2, \dots, C_k\}\), where \(C_i = \{v \in V(G) \mid \alpha(v) = i\}\). The color code of a vertex \(v \in V(G)\) with respect to \(\Pi\) is defined as the tuple \(c_{\Pi}(v) = (d(v, C_1), d(v, C_2), \dots, d(v, C_k))\), where \(d(v, C_i)\) represents the distance from \(v\) to the set \(C_i\). A proper \(k\)-coloring \(\alpha\) is called a locating \(k\)-coloring of \(G\) if \(\alpha\) induces a partition \(\Pi\) such that for any two distinct vertices \(u, v \in V(G)\), it holds that \(c_{\Pi}(u) \neq c_{\Pi}(v)\). The locating chromatic number of \(G\), denoted \(\chi_L(G)\), is the smallest \(k\) for which a locating \(k\)-coloring of \(G\) exists. In this paper, we establish a connection between the locating \(k\)-coloring of \(C_n(1,2,\dots,t) + K_m\) and the union of graphs \(\bigcup_{i=1}^p C_{n_i} + K_m\), leveraging properties of simple cycles in directed graphs. Using this connection, we determine the locating chromatic number of \(C_n(1,2,\dots,t) + K_m\) for \(t = 2\) and \(n \in [6, 28]\), as well as for \(t = 3\) and \(n \in [8, 24]\).

We characterize line digraphs of polytrees, including several of their well-known subclasses. For a given undirected tree, we characterize its orientations with weak line digraphs, and count the exact number. Furthermore, we find the minimum, maximum, and average sizes of these line digraphs. We provide an explicit formula for the number of weak components in line digraphs of polytrees in terms of the inner sources and sinks. Additionally, we count the average number of weak components in them among all orientations of a fixed tree. Finally, we propose an algorithm for finding weak components in line digraphs of polytrees.

Let \(k\ge 1\) be an integer. Let \(G=(V,E)\) be a connected graph with \(n\) vertices and \(m\) edges. Suppose fires break out at two adjacent vertices. In each round, a firefighter can protect \(k\) vertices, and then the fires spread to all unprotected neighbors. For \(uv\in E(G)\), let \(sn_{k}(uv)\) denote the maximum number of vertices the firefighter can save when fires break out at the ends of \(uv\). The \(k\)-edge surviving rate \(\rho'_k(G)\) of \(G\) is defined as the average proportion of vertices saved when the starting vertices of the fires are chosen uniformly at random over all eages, i.e., \(\rho'_k(G)=\sum\limits_{uv\in E(G)}sn_{k}(uv)/nm\). In particular, we write \(\rho'(G)=\rho'_1(G)\). For a given class of graphs \(\mathcal{G}\) and a constant \(\varepsilon>0\), we seek the minimum value \(k\) such that \(\rho'_k(G)>\varepsilon\) for all \(G\in \mathcal{G}\). In this paper, we prove that for Halin graphs, this minimum value is exactly 1. Specifically, every Halin graph \(G\) satisfies \(\rho'(G)> 1/12\).

A bipartite labeling of a tree of order \(n\) is a bijective function that identifies the vertices of \(T\) with the elements of \(\{0, 1, \dots, n-1\}\) in such a way that there exists an integer \(\lambda\) such that the set of labels on the stable sets of \(T\) are \(\{0,1, \dots, \lambda\}\) and {\(\lambda + 1, \lambda +2. \dots, n-1\}.\) The most restrictive and versatile bipartite labeling is the variety called \(\alpha\text{-labeling}\). In this work we present a new construction of \(\alpha\text{-labeled}\) trees where any two adjacent vertices of a path-like tree, or a similar caterpillar, can be amalgamated with selected vertices of two equivalent trees.

A set \(D\) of vertices of a graph \(G=(V_G,E_G)\) is a dominating set of \(G\) if every vertex in \(V_G-D\) is adjacent to at least one vertex in \(D\). The domination number of a graph \(G\), denoted by \(\gamma(G)\), is the cardinality of a smallest dominating set of \(G\). A subset \(D\subseteq V_G\) is called a certified dominating set of \(G\) if \(D\) is a dominating set of \(G\), and every vertex in \(D\) has either zero or at least two neighbours in \(V_G-D\). The cardinality of a smallest certified dominating set of \(G\) is called the certified domination number of \(G\), and it is denoted by \(\gamma_{\rm cer}(G)\). A vertex \(v\) of \(G\) is certified critical if \(\gamma_{\rm cer}(G -v) < \gamma_{\rm cer}(G)\), and a graph \(G\) is vertex certified domination critical or \(\gamma_{cer}\)–critical if the removal of any vertex reduces its certified domination number. In this paper, we give examples and properties of certified critical vertices and vertex certified domination critical graphs. As an example of an application of certified critical vertices, we give a constructive characterisation of trees for which the smaller partite set is a minimum certified dominating set.

In this paper, we consider Ramsey and Gallai-Ramsey numbers for a generalized fan \(F_{t,n}:=K_1+nK_t\) versus triangles. Besides providing some general lower bounds, our main results include the evaluations of \(r(F_{3,2}, K_3)=13\) and \(gr(F_{3,2}, K_3, K_3)=31\).

Seymour’s Second Neighborhood Conjecture (SSNC) asserts that every finite oriented graph has a vertex \(v\) whose second out-neighborhood is at least as large as its first out-neighborhood. Such a vertex is called a Seymour vertex. In this note, we introduce pseudo-Seymour set such that Seymour’s conjecture becomes: Every oriented graph has a singleton pseudo-Seymour set. We prove that any oriented graph has a pseudo-Seymour set \(S\) with \(|S|=2\). Furthermore, we show that there are pseudo-Seymour sets of any size at least 2. We define \(\rho\)-Seymour vertex where \(0 < \rho \leq 1\), and give an approach such that finding \(\rho=1\) is equivalent to the existence of Seymour vertex. Attempting to maximize its value, we prove that for any oriented graph with minimum out-degree \(\delta\), there is \(\rho=\frac{2}{3}(1+\frac{1}{2\delta})\).

In this paper, given a homeomorphism \(f\) of a compact metric space \(X\), we show that the set of all asymptotic average shadowable points of \(f\) is an open and invariant set and \(f\) has the asymptotic average shadowing property if and only if the set of all asymptotic average shadowable points of \(f\) is \(X\) if and only if any Borel probability measure \(\mu\) of \(X\) has the asymptotic average shadowing property.

This paper is concerned with the pullback attractors for the Kirchhoff type BBM equations defined on unbounded domains. Sobolev embeddings are invalid on unbounded domains. We obtain the pullback asymptotic compactness of such non-autonomous BBM equations by using the method of uniform tail-estimates.